Definition of Constants for Piezoceramic Materials [1 ed.] 9781613243312, 9781608763504

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Definition of Constants for Piezoceramic Materials, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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DEFINITION OF CONSTANTS FOR PIEZOCERAMIC MATERIALS

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DEFINITION OF CONSTANTS FOR PIEZOCERAMIC MATERIALS

VLADIMIR A. AKOPYAN ARKADY N. SOLOVIEV IVAN A. PARINOV AND

SERGEY N. SHEVTSOV

Nova Science Publishers, Inc. New York

Definition of Constants for Piezoceramic Materials, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Definition of constants for piezoceramic materials / Vladimir A. Akopyan ... [et al.]. p. cm. Includes bibliographical references and index. ISBN 978-1-61324-331-2 (eBook) 1. Piezoelectric ceramics. 2. Electronic ceramics. 3. Elasticity--Mathematical models. 4. Piezoelectricity--Mathematical models. 5. Piezoelectric devices. I. Akopyan, Vladimir A. TK7871.15.C4D44 2009 620.1'404297--dc22 2009041987

Published by Nova Science Publishers, Inc. Ô New York

Definition of Constants for Piezoceramic Materials, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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In memory of academician Iosiph Izrailevich Vorovich

Definition of Constants for Piezoceramic Materials, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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CONTENTS Preface Chapter 1

Introduction

1

Chapter 2

Boundary Problems for Piezoelectrics and Mathematical Models of Electro-Elasticity

9

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xi

Methods for Definition of Piezoelectric Material Constants

49

Identification of Oscillation Modes and Determination of Full Matrix of the Compatible Piezoelectric Material Constants

59

Chapter 5

Definition of Piezomodule d33

105

Chapter 6

Definition of Elastic Constants for Piezoceramics and their Temperature Characteristics by Using Method of Bending-Torsion Oscillations of the Cantilever Beam

119

Specialized Finite-Element Complex ACELAN

159

Appendix A References

189

Index

199

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PREFACE The monograph consists of the investigation results obtained on the base of new theoretically well-grounded method for definition of total set of the compatible material constants for different piezoceramics. With this aim, there are carried out theoretical and experimental analysis, and also identified various oscillation modes of electroelastic bars and plates. Moreover, the physical and geometrical restrictions on the sample form are stated by using modified method of definition of the piezomodule d33 in quasistatic regime. These limitations allow ones to calculate the valid values of d33 for various piezoceramic families. By this, the boundaries of application of the above methods are also discussed. As a rule, the piezoelements of simple geometric forms apply into sensors, piezoelectric resonators, filters and piezoelectric transducers which are widely used in different scientific and technical goods. The computations of characteristics of these piezotransducers are based on known mathematical models and standard methods [64, 69, 116]. A difference of the calculated and experimental values not exceeds usually of permissible limits and takes into account during engineering finish off of piezotransducers and in technology their processing [59]. However, the majority of applied transducers permit very narrow range of displacements and forces of piezoelements lesser than 100 − 200 μm, that is connected with high brittleness and small possible deformations of piezoceramic materials. At the same time, new devices of intellectual technique, created in the last years, demand the piezoelements with wide diapasons of displacements and forces exceeding the above limits. Today, are known different ways for significant increase of deformations of the piezoelements. With this view, the multilayer piezoelements and elements with complex geometry have been investigated in the last years. Moreover, there

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Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

are known the investigation results of piezotransducers consisting of piezoelectric components. Often, the calculations their characteristics on the base of traditional methods lead to the significant errors. The presented in the book results of identification of the various oscillation modes for bars and plates, and also the method and algorithm of definition of total set of the compatible material constants may significantly help in development of the computation methods for characteristics of piezocomposite materials for force piezoactuators. One of the approaches in order to achieve this aim consists in that the some important for practice forms of maximum displacements could be created by using the stiffness matrices of composite based on the calculation algorithm of total set of the compatible piezoceramic material constants [19]. Moreover, above modified algorithm may be used for research and development of new composites, in particular polymer-composite materials [12, 13, 122]. The book presents totally remade Russian edition of the book [21]. The monograph is addressed to students, post-graduate students and specialists, taking part in the development, preparation and researching new materials and devices on their base. Authors have pleasure to thank Russian Foundation for Basic Research, Russian Department of Education and Science grants which during last decade have rendered considerable financial supporting and promoted to publish this book. We are also grateful to colleagues and nearest scientific workers, who have directly or indirectly contributed to the book. In particular, we wish to thank A. V. Belocon, V. Z. Borodin, A. V. Nasedkin, E. V. Rozhkov, A. O. Vatulyan, Yu. N. Zakharov. Corrections and proposals of the book readers will be considered with thanks. They could be presented by E-mail: [email protected].

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Chapter 1

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INTRODUCTION Acoustic-electronics is one of the priority directions of modern science and technology. The various acoustic and piezoelectric devices have been researched and developed. They have widely applied in radio-electronics, hydro-acoustics, instrument-making industry and non-destructive control systems. There are numerous monographs and overviews devoted to devices based on the piezoceramic materials (PCM), their properties and behavior under different loading. Among them, there should be mentioned the works of W. Mason, Ye. T. Smazhevskaya, N. B. Fel’dman, V. Z. Borodin, Ye. G. Fesenko, O. P. Kramarov, R. Y. Kazhis, R. G. Dzhagupov, A. A. Yerofeev, A. A. Ananieva, V. V. Lavrinenko, et al. [22, 44, 45, 51, 54, 74, 83, 89, 126]. Initially, the piezoceramics had been applied in hydro-acoustics, namely in hydrophones to receive hydro-acoustical signals by acoustic stations using acoustic antenna in the form of cylinder of diameter 2 − 4.5 m, with some hundreds of piezoelements located on its surface (e. g. the antenna of hydrolocator AN/SQS-26 of WSF USA includes 576 piezoelements). Further fast development of piezoelectronics led to creation of piezotransducers, piezoengines, delaying lines and so on [57]. Obviously, the various types of the PCM devices work in wide diapason of pressures and temperatures. Therefore, in order to achieve a reliable work and select optimal conditions of functioning of the PCM devices, it is necessary to investigate the total set of material properties. This

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theme has been in the centre of many theoretical and experimental studies [27, 46, 47, 60, 71, 79, 82, 84, 88, 99, 105, 108, 109, 111, 123, 124, 131, 133, 141]. Among more significant contributions devoted to the mathematical models of electro-elasticity, it is necessary to mention whole set of fundamental monographs and papers of B. S. Aronov, R. Bechman, A. V. Belocon, V. T. Grinchenko, R. Holland, V. L. Karlash, B. A. Kudryavtsev, V. V. Meleshko, A. V. Nasedkin, V. Z. Parton, N. A. Senik, N. A. Shul’ga, H. F. Tiersten, A. F. Ulitko, Yu. A. Ustinov, A. O. Vatulyan, I. I. Vorovich, et al. [23, 29, 35, 36, 57, 61, 62, 63, 66, 92, 104, 118, 129, 137]. These investigations have been devoted to the different statements, theoretical and numerical methods of solution above problems of electro-elasticity. In engineering, the linear theory of electro-elasticity is usually applied in which the PCM properties are described by using a set of elastic and piezoelectric modules and dielectric permeability. In this case, it is assumed that the piezoceramic solid presents homogeneous medium. The problem to define the full set of the piezoceramic material compatible constants, characterizing physical properties of anisotropic materials with transverse-isotropic symmetry (in particular, piezoceramic materials), has not yet solved, in total with necessary degree of reliability in definition of the material constants. Nevertheless, today in practice, it is used a standard test method of ‘resonance-antiresonance’ (MRA), based on constitutive equations of linear theory of the electro-elasticity [64, 69, 116]. Generally, this method allows one to compute a full set of the piezoceramic material non-compatible constants. Obviously, these constants describe the piezoceramic properties which are differed from material constants of initial materials. The values of the constants calculated by using MRA are reliable for corresponding types of piezotransducers of a geometry that is coincident with the shape of the samples used for computation of the given constant. However, their values include errors due to the difference of the sample properties. Due to the constants of the piezoceramic materials are defined investigating properties of the samples processed by using various technological regimes, they can not to be considered as compatible. It is followed from the experimental data witnessing on significant heterogeneity of physical properties of the piezoceramic materials caused by sintering and polarization of piezoelements with various geometries. The modern methods to define the compatible constants allow one to compute a set of constants, as rule for monocrystals [84, 138]. In Reference [40], the set of the constants for piezocrystals has been found by using the acoustic method and four types of the samples, but a compatibility of the constants has not been achieved. In References [46, 71, 84, 124], the total set of the constants for

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Introduction

3

piezoceramics has been found by using the ultrasound method and four types of the same material. A total compatibility of the constants has not been obtained, too. In Reference [84], it has been assumed a method for definition of the total set of the piezoceramic constants by using MRA and four types of samples cut out from one initial sample. However, in one of these sample the old electrodes have been removed from the sample faces and new electrodes created on other faces. Obviously, this operation introduced an additional error violating a total compatibility of the material constants. It is also known the paper [138], in which the immerse method is proposed to state the total set of piezoceramic constants by using phase velocities of the differently polarized samples at the frequency of 30 MHz. In this case, the total compatibility of the material constants has not been ensured. Therefore, the constants measured at high frequencies have not been always coincided with ones defined into kHz-frequency band of work of the majority piezotransducers. This result has been obtained, earlier [131]. Moreover, the known methods (in particular, MRA and impulse method) not allow one to define the constants into wide range of elevated temperatures with errors that are admissible for applications. In the same time, many transducers work in the zone of elevated temperatures, e. g. into engines of aircraft and energetic aggregates. Obviously, the research and development of the methods and algorithms to define the full set of the piezoceramic compatible constants are very important in order to modernize the physical models of piezoelectricity, create well-founded techniques of sintering and polarization of the piezoceramic precursor samples and also to state numerical methods for new types of piezotransducers. In particular, without definition of total set of the compatible piezoceramic constants, it is very difficult to solve a whole set of important scientific and technical problems, namely: •

•

Select the pressing, sintering and polarization regimes of piezoceramics allowing one to obtain maximal factors of electromechanical coupling and limit high sensitivity in piezoelements into given ranges of work frequencies and temperatures. Develop multi-resonance piezotransducers of beam and disk type with high sensitivity on some resonance frequencies of different oscillation modes, generated by using the same piezoelement. These transducers allow one to find the defects with preliminary given sizes. This circumstance is very important for the technical structural health monitoring.

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•

Create piezocomposite elements of intelligent structures (girders of satellite antennas and optic mirrors, helicopter rotor blade, etc.), including piezoelements, formed into composite, and working in regime of maximal amplitudes of the different oscillation modes, generated by using the same type of a sample. The research and modeling these are carried out after definition of the set of compatible piezoceramic and composite constants that describe electrical and physical characteristics of composite element with the given geometric and physical configuration.

The proposed in monograph [21] method to define full matrix of the piezoceramic material compatible constants, based on the theoretical and experimental identification of the various oscillation mode shapes in thin rectangular prism together with the method of bending-torsion oscillations of the beam piezoelements from polarized and non-polarized ceramics under elevated temperatures, allow one to improve the known methods [10, 11] and propose the control systems for modeling of damping processes of construction element oscillations by using piezoactuators [12, 18, 20, 122]. The methods, presented in the monograph, have been patented in Russia [7, 76]. The book is divided into six chapters and one appendix. Chapter 2 is devoted to statement of boundary problems for piezoelectric medium on the base of four equivalent forms of the constitutive equations for linear piezoceramic material, obtained from set of thermodynamic potentials, namely internal energy, electric and mechanical enthalpy [44, 76, 126], and also some mathematical models of electro-elasticity. Paragraph 2.1 presents constitutive equations of electro-elastic media, describing electro-elastic properties of piezoceramics. The equations are obtained for case of selection as independent variables the mechanical stresses and electric field intensity written in tensor forms. These equations are transformed into matrix form after substitution of indices. As a result, it has been obtained the elasto-piezoelectric matrix of the piezoceramic constants [57, 126]. The matrix components include ten independent constants that are sufficient for description of the piezoceramic properties. The piezoelectric materials to be transverse-isotropic medium of crystallographic class 6mm with polar axis of elastic symmetry of the infinite order (the conditions of transverse-isotropic symmetry are fulfilled). Therefore, obviously that for description of electromechanical properties of the piezoceramics, there are sufficient to know five independent elastic constants, namely two dielectric permeability and three piezoelectric constants. In Paragraphs 2.2 − 2.5, the mathematical models of the electro-elasticity problems, used in experiments, are present. In particular, the

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Introduction

5

problem on longitudinal oscillations of piezoceramic rectangular prismatic beam is solved in paragraph 2.2. It is shown, that there are resonances of four types observable in a spectrum. One of them corresponds to the low-frequency mode of beam with electric field transverse to oscillation displacement (BLt), and second − to the mean-frequency mode of rectangular prism with electric field parallel to oscillation displacement (PMp). In this case, an identification of these oscillation modes is possible at definite geometrical sizes of piezoelements. The estimation of the result validity in the above mode identification has been carried out by using computer simulation of the frequency response for vibration amplitude of beam and plate − amplitude-frequency characteristic (AFC) of the samples, and construction of forms their oscillations based on the realization of computer results obtained by using the finite-element program complex ACELAN [35, 121]. The numerical data have confirmed a high degree of validity of the results measured. In Paragraphs 2.3 − 2.5, the solutions are obtained for other electroelasticity problems. In particular, there are considered the problems on longitudinal oscillations of a beam under longitudinal electric field, on shear oscillations excited in plate with the depth polarization, and also the problems on planar oscillations in thin plate polarized in depth. Moreover, there are obtained relationships for calculations of total set of the piezoceramic material constants. In Chapter 3, the known experimental methods determining elastic and piezoelectric constants of piezoceramics are analyzed briefly. Paragraph 3.1 describes the test methods for elastic and piezoelectric constants defined in dynamics. A systematization of these methods is carried out on the base of the loading frequency of sample considered. The comparative analysis of these methods has showed that by using MRA [64, 69, 116], it may be estimated a total set of the piezoceramic constants that are incompatible due to the measurement of the resonance frequencies by using three samples with different geometry and polarization degree. In lesser degree this conclusion is related to the impulse method [139, 143] and the method of phase velocities [91], in which the measurements of velocities are fulfilled by using two various samples. This procedure again does not ensure total compatibility of the constants. Moreover, these methods allow ones to calculate the constants describing properties of the piezoelements with various configurations, differing from properties of the piezoceramic material considered. Then, it is shown that by using MRA, it is impossible to define the constants of non-polarized ceramics, but the temperature dependencies of the constants at the elevated temperatures may be calculated with high error. In order to exclude demerits above mentioned methods the authors developed modernized method to determine a total matrix of the compatible material dynamic constants for piezoceramics. This method has been completed

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by new method of definition of the elastic modules for polarized and nonpolarized ceramics at elevated temperatures by using the method of bendingtorsion oscillations [11]. In Paragraph 3.2, the known measuring methods for piezomodules of piezoceramics are analyzed. This analysis shows that quasi-static methods allow ones to estimate piezoconstants with lesser errors by comparing with other methods. Chapter 4 is devoted to the identification problem of oscillation modes and statement the corresponding total matrix of the compatible material piezoceramic constants. Paragraph 4.1 discusses research results on identification of the different oscillation modes obtained for piezoceramic rectangular prismatic beam including the longitudinal and depth modes for transverse-polarized beam. There are present the test results of estimation of the resonance frequencies for these modes, theoretical analysis of AFCs and oscillation forms of the beam, and also comparative data of theory and experiments. In Paragraph 4.2, the method and estimation algorithm of total matrix of the compatible material piezoceramic constants are described, and the measuring device for their realization considered. Moreover, the measuring results for total matrix of the material constants are present for piezoceramics PCR-1, PCD-124, PZTB-31. The obtained errors are estimated, the effect of microstructure parameters of piezoceramics on elastic properties stated, and a comparative analysis of the obtained values with known data carried out. Chapter 5 presents a modernized method and device to calculate the piezoelectric module d33 in quasi-static regime. First, the relationships for calculation of d33 are stated, the test method and measuring results are described with following estimation their reliability. Then, the error calculation is carried out for piezomodule. Chapter 6 discusses the problem of the forced bending oscillations of the piezoceramic cantilever beam with additive mass. The general solution obtained is reduced to simple relationship used for calculation of elastic modules of the piezoceramic materials into thermal range of 300 − 380 К. Then, it is present the method of definition of the technical elastic constants by using the experimental method of bending-torsion oscillations. This method allows one to calculate the technical modules of elasticity and shear for polarized and non-polarized

1

PCR-1 is the Rostov Piezoceramic-1 of the high-sensitive piezoelectric composition of the leadzirconate-titanate (PZT) family with rhombohedronic structure near to boundary of the morphotropic region [54, 59], PCD-124 is the Donetsk piezoceramic-124 of composition of the lead-zirconate-titanate-strontium, PZTB-3 is the piezoceramic of the composition lead-zirconatetitanate-barium.

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piezoceramics (PZTB-3 and PZTNB-12) with different ferrohardness and obtain representative data. Moreover, there are present the numerical results of effects of the stress-strain state and thickness of the electrode covering of piezoelements on the elastic properties of piezoceramics. Then, it is found relationship between the technical elastic modules of polarized and non-polarized piezoceramics into range of elevated temperatures. Finally, the analyses of validity of the calculated elastic constants and the computation of their errors are carried out. Appendix A presents in detail the specialized finite-element complex ACELAN developed, in particular to investigate piezoceramic materials and devices. First, the continuum models of elastic, electroelastic and acoustic media are discussed. Then, the corresponding finite-element models are constructed. Moreover, it is stated a procedure of numerical solution of the obtained system of the ordinary differential equations for the finite-element model. The examples relate to the problems of steady oscillations and modal analysis.

2

PZTNB-1 is the piezoceramic of the composition lead-zirconate-titanate and sodium-bismuth oxide [59].

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Chapter 2

BOUNDARY PROBLEMS FOR PIEZOELECTRICS AND MATHEMATICAL MODELS OF ELECTRO-ELASTICITY

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2.1. CONSTITUTIVE EQUATIONS OF ELECTRO-ELASTIC MEDIA The polarization of dielectrics is the phenomenon connected with formation of the coupled charged particles in the dielectrics placed into electric field. These particles are capable under field to displace in short distances limited by their interaction with crystalline lattice. The polarization vector P with components

Pi

(i = 1, 2, 3) could be found from the equation [104]

ρ ch = − where

∂Pi , ∂xi

(2.1)

ρ ch is the volume density of the connected charges in dielectrics and xi

are the spatial coordinates. Here and below, it is assumed a summing on repeated indexes. The vector P is called as the vector of dielectric polarization. One defines the volume density of the connected charges and also a surface density of the connected charges distributed on surface of polarized dielectric sample. The physical meaning of the polarization vector is found by the dipole momentum M

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of all connected charges in the dielectric sample with volume V. Its components are found in the form

M i = ∫ xi ρ ch dV .

(2.2)

V

Then Equations (2.1) and (2.2) lead to the next relationship

M i = ∫ Pi dV .

(2.3)

V

This equation shows that whole dipole momentum M of all connected charges in the dielectric sample is found by integrating of the polarization vector P on the dielectric volume V. Therefore, in each point of the dielectric sample we have

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Pi =

dM i . dV

(2.4)

The polarization vector P is the electric dipole momentum of the dielectric volume unit. It is well known, that the polarization vector P in ferroelectric materials may be differ from zero at the electric field intensity E = 0. This polarization is called as spontaneous, and materials with this polarization to be pyroelectrics. The crystals, demonstrating the spontaneous polarization only in definite temperature range, are related to the ferroelectric kind the spontaneous dipole momentum of which is stated by the displacement of the ion sublattices. It has been proved experimentally, that the spontaneous dipole momentum in ferroelectrics exists only in definite temperature range of stable ferroelectric phase. In this case, both limit values of the temperature for this range are called by upper and lower Curie temperatures [89]. Moreover, it is also known, that the ions of the crystalline lattice of ferroelectrics causing an initiation of the spontaneous polarization could be displaced by mechanical stresses. This phenomenon signs that the mechanical loading of ferroelectric crystal leads to initiation of the electric polarization, and it is called the piezoelectric effect. The crystals demonstrating this effect are called piezoelectrics. The piezoelectric materials present a polycrystalline solid solution of monocrystals with the polarization vector oriented by strong external electric field.

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Boundary Problems for Piezoelectrics and Mathematical Models …

11

The straight piezoelectric effect is displayed in initiation on crystal faces the electric charge Q that is proportional to applied mechanical stress σ. In the same time, the reverse piezoelectric effect assumes an initiation on the crystal faces the mechanical deformation under influence of electric field with the electric field intensity E. The straight piezoelectric effect is observed only for crystals of definite symmetry groups under mechanical loading. Usually, it is assumed in the theory of piezoelectric materials that at the zero electric field intensity, the polarization vector of piezoelectrics is linearly related with the components of the mechanical stress tensor

σ kl

(k, l = 1, 2, 3)

Pi = d ikl σ kl , where

d ikl

(2.5)

are the piezoelectric modules forming a tensor of third rank.

The mathematical equation of the reverse piezoelectric effect describes linear relationship between electric field intensity Ei and strain tensor ε ij for crystal without mechanical loading ( σ ij

= 0)

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ε ij = d kij Ek .

(2.6)

Due to the symmetry of the strain tensor components ( ε ij followed from Equation (2.6) that the tensor

d kij

= ε ji ),

it is

is symmetric on pair of last

indexes. Owing to this symmetry the number of independent components of the tensor

d kij

is equal to 18. The following decreasing of this number is connected

with the crystal symmetry. In the case of existing symmetry center, all components of the tensor

d kij

are equal to zero i. e. the crystal has not

piezoelectric properties [104]. The piezoceramic materials are present by transverse-isotropic medium in each point of which there is a polar axis of elastic symmetry of infinite order. In the case of piezoelectrics demonstrating the transtropic symmetry, some components of the tensor

d kij

are not equal zero [89], that proves presence the

piezoelectric properties of these materials. Taking into account these circumstances and using as thermodynamic potentials the function of internal energy, the linear equations of piezoelectric state have been obtained in

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References [89, 104] at small changes of the variables ε ij , Dm , E m and into framework of the thermal treatment. By neglecting the heat-exchange at the elastic oscillations of the piezoelectric media, the adiabatic equation of the state have been obtained in the form [89, 104] D σ ij = cijkl ε kl − hijm Dm ,

(2.7)

ε E m = −hmij ε ij + β mk Dk ,

where

⎛ ∂σ ij D c ijkl = ⎜⎜ ⎝ ∂ε kl

⎞ ⎟⎟ ⎠D

is the tensor of elastic modules at the constant electric

induction Dm = const or Dm = 0;

piezoelectric constants;

⎛ ∂σ ij hijm = ⎜⎜ ⎝ ∂D k

⎛ ∂E ⎞

ε = ⎜⎜ m ⎟⎟ β mk ⎝ ∂Dk ⎠ ε

⎞ ⎟⎟ ⎠ε

is the tensor of adiabatic

are the adiabatic dielectric constants at

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the constant deformations to being the components of the tensor of dielectric permeability. Into Equations (2.7) the strain

ε kl

and electric induction Dm are the

independent values. The analogous to Equation (2.7) equations could be obtained by using other thermodynamic potentials. In particular, in the case of independent variables of strain ε kl , electric field intensity Ei and temperature T (with dependent variables

σ ij , Di

and entropy S), the Gibbs electric function is the

corresponding thermodynamic potential. By using this function the next constitutive equations have been obtained in absence of heat-exchange [89, 104]: E T σ ij = cijkl ε kl − eijm Em ,

ε ,T T Dm = эmk E k + emij ε ij ,

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(2.8)

Boundary Problems for Piezoelectrics and Mathematical Models …

where

13

⎛ ∂σ ij ⎞ T ⎟⎟ is the tensor of piezoelectric constants defining a eijm = −⎜⎜ ∂ E ⎝ m ⎠ε

change of the mechanical stresses due to electric field;

⎛ ∂D ⎞ ε ,T эmk = ⎜⎜ m ⎟⎟ ⎝ ∂E k ⎠ ε

is the

tensor of dielectric permeability at the constant deformations. Another form of more applied constitutive equations describing electro-elastic properties of piezocrystals has been stated in the case of selection as independent variables the stresses

σ kl

and electric field intensity Ek in the form [89, 104]:

E ε ij = sijkl σ kl + d kij E k ,

(2.9)

Di = d ijk σ jk + эijσ E j , where

E are the isothermal factors of elastic compliance; d kij are the s ijkl

σ

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piezoelectric modules; эij is the tensor of dielectric permeability. The constitutive Equations (2.7) − (2.9) written in the tensor form, it is handy to rewrite in a matrix form. With this aim, it is necessary to carry out the index substitution in which last two indexes are substituted in according with the next scheme: tensor indices jk: matrix indices m:

11 1

22 2

33 3

23 4

32 4

31 5

13 5

12 6

21 6

and also we state the next equalities:

d im = d ijk at the m ≤ 3,

d im = 2d ijk at the m ≥ 4.

(2.10)

Two last indices (jk) in the tensor presentation corresponds to indices of stress components, and therefore the stress components in the matrix form are found as

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Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

⎡T11 T12 ⎢T T ⎢ 21 22 ⎢⎣T31 T32

⎛ T1 ⎞ ⎜ ⎟ ⎜T ⎟ T13 ⎤ ⎜ 2 ⎟ T T23 ⎥ → ⎜ 3 ⎟ . ⎥ ⎜T ⎟ T33 ⎥⎦ ⎜ 4 ⎟ T ⎜ 5⎟ ⎜T ⎟ ⎝ 6⎠

(2.11)

Moreover, due to symmetry of the tensor

sijkl = sαβ

(i, j, k, l = 1, 2, 3; α‚ β = 1, 2,…,6).

Hence, the factors of elastic compliance elastic modules

cαβ

sαβ

(2.12) (α‚ β = 1, 2, …, 6) as and

form square matrix (6×6). Analogously, we obtain the

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matrix of piezoelectric constants [104]. By taking into account the designations introduced the constitutive equations for isothermal conditions may be written as

ε i = sijE T j + d mi E m ,

(2.13)

T Dm = d mi Ti + эmk Em .

The equations of piezoeffect for adiabatic conditions (2.7) differ from Equations (2.13) due to the adiabatic constants

T , d mi s ijE , эmk

in these

conditions substitute corresponding isothermal ones. In the dependence on selection of independent variables, the constitutive equations could be written in four different forms [104]. The number of independent factors of Equations (2.13), which characterize elastic and electric properties of medium, depends on the crystalline symmetry [89]. In the case of piezoelectric material being transverse-isotropic medium (class 6mm) [89, 104], there is a polar axis of elastic symmetry of infinite order in each point of the solid, but all directions in the planes perpendicular to this axis are

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Boundary Problems for Piezoelectrics and Mathematical Models …

15

equivalent in relation to elastic properties. Due to the elasto-piezoelectric matrix of piezoceramic constants has form [57, 126]:

s11 s12 s13

s12 s11 s13

s13 s13 s33

0 0 0 0 0 d 31

0 0 0 0 0 d 31

0 0 0 0 0 d 33

0 0 0 s 44 0 0 0 d15 0

0 0 0 0 s 44 0 d15

0 0 0 0 0 0 0 0 d15 0 2( s11 − s12 ) 0 э11 0

0 0

Here d kl are the piezomodules and

0 0

эmn

0 0

0 0 0 d15 0 0 0 э11 0

d 31 d 31 d 33 0 (2.14) 0 0 0 0 э33

are the components of the dielectric

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permeability tensor. Then, we obtain from the condition of transverse-isotropic symmetry [57, 89, 126]

s13 = s 23 , s11 = s22 , s44 = s55 = s66 ,

d 31 = d 32 , d 24 = d15 , э11 = э 22 .

(2.15)

Matrix (2.14) shows that in other to calculate mechanical, dielectric and piezoelectric properties of piezoelectric and another transverse-isotropic media, it is sufficient to have five independent elastic constants, namely two dielectric permeability and three piezoelectric constants. The constitutive equations for piezoceramics polarized in axis x3 through matrix designation have forms

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Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

σ 11 = c11E ε 11 + c12E ε 22 + c13E ε 33 − e31 E3 , σ 22 = c12E ε 11 + c11E ε 22 + c13E ε 33 − e31 E3 , σ 33 = c13E (ε 11 + ε 22 ) + c33E ε 33 − e33 E3 , σ 23 = 2c44E ε 23 − e15 E2 , σ 13 = 2c44E ε 13 − e15 E1 , σ 12 = (c11E − c12E )ε 12 ,

(2.16)

D1 = э11ε E1 + 2e15ε 13 , D2 = э11ε E2 + 2e15ε 23 , ε D3 = э33 E3 + e31 (ε 11 + ε 22 ) + e33ε 33 ,

where

1 ⎛ ∂u

∂u j ⎞

⎟ ε ij = ⎜⎜ i + 2 ⎝ ∂x j ∂xi ⎟⎠ ∂ϕ is the electric

Ei = −

∂xi

are the components of the small strain tensor;

field intensity;

ϕ

is the electric potential;

E E are the elastic modules measured in electric field with c11E , c12E , c13E , c33 , c 44

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constant (or zero) value;

ε ε e31 , e33 , e15 are the piezoelectric constants; э11 , э33

are the components of the dielectric permeability at the constant strains. Analogously, the constitutive equations for piezoceramics could be written in matrix form in which the mechanical stresses and the electric field intensity (see Equations (2.13)) are the independent variables

ε11 = s11E σ 11 + s12E σ 22 + s13E σ 33 + d 31 E 3 ,

ε 22 = s12E σ 11 + s11E σ 22 + s13E σ 33 + d 31 E 3 , ε 33 = s13E (σ 11 + σ 22 ) + s33E σ 33 + d 33 E 3 ,

ε 23 = s 44E σ 23 + d15 E 2 ; ε 13 = s 44E σ 13 + d15 E1 , T D1 = э11 E1 + d15σ 13 ; D2 = э11T E 2 + d15σ 23 , T D3 = э33 E 3 + d 31 (σ 11 + σ 22 ) + d 33σ 33 ,

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(2.17)

Boundary Problems for Piezoelectrics and Mathematical Models … where

E E s11E , s12E , s13E , s 33 , s 44

17

are the factors of elastic compliance at the

electric field intensity of constant (or zero) value; d 31 , d 33 , d15 are the Т Т are the components of the dielectric permeability at the piezomodules; э11 , э33

ε11 , ε 22 , ε 33 , ε 23 , ε 13 , ε 12 are the components of the strain tensor; σ 11 , σ 22 , σ 33 , σ 23 , σ 13 , σ 12 are the

constant or zero mechanical stresses;

components of the stress tensor. It may be shown [104] that it is assumed to neglect the magnetic effects in the problems of propagation of the electro-acoustic waves in piezotransducers. In this case, we obtain the quasistatic approximation for electric field as

ε ijk

∂E j ∂x k

=−

∂Bi ≈ 0 , ∂Di = 0 , ∂t ∂xi

(2.18)

Bi = μH i is the vector of magnetic induction; E j , H j are the vectors of the electric and magnetic field intensities; μ is the magnetic permeability.

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where

Taking into account these equations, the complete system of the constitutive equations for linear piezoelectric medium without thermal effects may be obtained by substitution of Equations (2.16) into motion equations [89]:

∂σ ij

∂ 2ui =ρ 2 ∂x j ∂t

(2.19)

and into the electrostatic equation

∂Di = 0. ∂xi

(2.20)

The system of linear equations of the electro-elasticity for piezoelectric medium is stated after substitution of a relationship for the equations obtained for

Ei into Equation (2.7) and

σ ij and D j into the movement equations (2.19) and

the electrostatic equation (2.20). In the orthogonal system of coordinates

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Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

( x1 , x 2 , x3 ), this system of the constitutive equations for the materials with transverse isotropy (hexagonal type of symmetry 6mm with the axis of the symmetry of 6th order coinciding with the axis x3), and taking into account twoindex designations for the elastic modules cαβ and piezomodules d im has been obtained in Reference [104]. By retaining figure indices at the material constants, we obtain 2 2 ∂ 2u3 ∂ 2 u1 1 E E ∂ u1 E E E ∂ u1 + c 44 + (c13 + c 44 ) + (c11 − c12 ) 2 + c ∂x1∂x3 2 ∂x12 ∂x32 ∂x 2 E 11

∂ 2 u1 ∂ 2u2 1 E ∂ 2ϕ E =ρ 2 , + (c11 + c12 ) + (e31 + e15 ) 2 ∂x1∂x3 ∂x1∂x 2 ∂t (2.21) 2 ∂ 2 u1 ∂ 2u ∂ 2u2 1 E 1 E ∂ u2 c + (c11E − c12E ) 22 + c11E + + (c11 + c12E ) 44 ∂x1∂x2 2 2 ∂x1 ∂x22 ∂x32

∂ 2 u3 ∂ 2u2 ∂ 2ϕ + (c + c ) + (e31 + e15 ) =ρ 2 , ∂x2 ∂x3 ∂x2 ∂x3 ∂t

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E 13

E 44

(2.22)

E (c13E + c 44 )

2 ∂ 2u3 ⎞ E ∂ 2u3 ∂ 2 u1 E ⎛ ∂ u3 ⎟ + c33 ⎜ 2 + + + c 44 2 ⎟ 2 ⎜ ∂x ∂x1∂x3 x x ∂ ∂ 2 ⎠ 3 ⎝ 1

⎛ ∂ 2ϕ ∂ 2ϕ ⎞ ∂ 2u3 ∂ 2u 2 ∂ 2ϕ ⎜ ⎟ + (c + c ) + e15 ⎜ 2 + 2 ⎟ + e33 2 = ρ 2 , ∂x 2 ∂x3 ∂t ∂x3 ⎝ ∂x1 ∂x 2 ⎠ E 44

E 13

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(2.23)

Boundary Problems for Piezoelectrics and Mathematical Models …

(e31 + e15 )

⎛ ∂ 2u ∂ 2u3 ⎞ ∂ 2u3 ∂ 2 u1 ⎟ e + e15 ⎜⎜ 23 + + + 33 2 ⎟ 2 ∂x1∂x3 x x x ∂ ∂ ∂ 1 2 3 ⎝ ⎠

∂ 2u2 ∂ 2ϕ ∂ 2ϕ ∂ 2ϕ + (e31 + e15 ) − э11 2 − э11 2 − э33 2 = 0. ∂x 2 ∂x3 ∂x1 ∂x 2 ∂x3

19

(2.24)

Consider the boundary conditions for the system of Equations (2.21) – (2.24). Usually in the problems of electro-elasticity, the mechanical and electric boundary conditions are separated. The mechanical field components are found, as rule through the vector of mechanical stresses T or the displacement vector u, and also in the form their combination for each concrete problem. The formulations of electric boundary conditions for typical cases of loading of the piezoelements are present in References [89, 104, 124, 134]. Let us piezoelectric medium with volume V is bounded by surface S. Then the boundary conditions are reduced to usual relationships of the Solid Mechanics, namely

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σ ij n j = ti

in Sσ,

ui = Vi

in Su,

(2.25)

where nj is the unit normal vector to the surface S; Sσ, Su are two parts of the surface S in which the external loading ti and displacements ui are respectively given. If the electric boundary conditions are connected with method of supply of the electric energy to polarized sample with electrodes brought in the surface Sφ, presenting a part of the total surface S, and when voltage at the electrodes is found by the electric potential ± V0 e iωt , then the electric boundary condition in Sφ has the form

ϕ

Sϕ

= ± V 0 e i ωt .

(2.26)

If the value of the electric potential is unknown in electrodes, and the electric current is given in circuit, then the boundary condition in electrodes of Sφ for piezoceramic sample will be found as

ϕ

Sϕ

= ± V (t ) .

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(2.27)

20

Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al. In the last case, the unknown value of electric potential V (t ) at the

electrodes is found through the known value of the electric current I (t ) in the external circuit. It is known [134] that this dependence may be written as

∂ ( n⋅D )dS = − I (t ) , ∂t S∫ϕ

(2.28)

where Sφ is the sample surface with electrodes. The potential V (t ) is not

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evidently included into this equation through a solution of the boundary problem of electro-elasticity. In the general case, if the surface of piezoelectric solid bounds with external medium, for example with air, then it is necessary to add the Maxwell equations for this medium to Equations (2.21) – (2.24) by taking into account a continuity of the tangential component of the electric field intensity vector E and the normal component of the electric induction vector D in the surface S (in the case of absence of the free electric charges at the interface between above two media). The electric boundary conditions in the surface parts of the piezoelectric solid without electrodes SD are formulated as n⋅D = n⋅D(0), et ⋅E = et ⋅E(0).

(2.29)

where et is the unit tangential vector to the surface S. In the case of the applied problems, the external problem of electrostatics is not solved, and the first Equation (2.29) reduces to the condition n⋅D | S D =

0.

2.2. ELECTRO-ELASTICITY PROBLEM ON LONGITUDINAL OSCILLATIONS OF TRANSVERSE-POLARIZED BEAM Consider a statement of the problem on oscillations of preliminary polarized beam with rectangular cross-section having in the orthogonal system of coordinates Ox1x2x3 the volume V = {−l ≤ х1 ≤ 0; 0 ≤ х2 ≤ b; −а ≤ х3 ≤ 0}. The direction of the electric field of the preliminary polarization coincides with the positive direction of axis Oх3 (see Figure 2.1).

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Boundary Problems for Piezoelectrics and Mathematical Models …

21

Figure 2.1. Polarized beam with the electrodes which are shaded.

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In the case, when the mechanical stresses and electric field intensity are the independent variables, the constitutive equations have form of Equations (2.17). The dependences between displacements and strains are determined by Cauchy equations [133]. The motion equations in the case of neglecting the volume forces are found by well-known Equations (2.19). Electric field in piezoceramic may be described by the electrostatic Equation (2.20) of dielectrics. Then, state boundary conditions for the system of Equations (2.21) – (2.24). Consider the boundary conditions for the beam faces. In the case, when the faces х1 = −l, х1 = 0, х2 = 0, х2 = b contact with air, we have the next boundary conditions at these beam faces: Dn = n⋅D = 0,

(2.30)

where п is the unit normal vector to the corresponding face. The beam faces х3 = 0 and х3 = − а have electrode covering and in them the electric potential is given as

ϕ = ± V0 e iωt .

(2.31)

where t is the time; ω is the cyclic frequency of oscillations. Below, we assume that the faces of the piezoceramic beam are free from mechanical stresses. For the faces х1 = −l, х1 = 0, the boundary conditions are written as

σ 11 = 0, σ 13 = 0, σ 12 = 0

at the

σ 11 = 0, σ 13 = 0, σ 12 = 0

at the x1 = −l , 0 ≤ x2 ≤ b, − a ≤ x3 ≤ 0.

x1 = 0, 0 ≤ x2 ≤ b, − a ≤ x3 ≤ 0, (2.32)

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σ 22 = 0, σ 23 = 0, σ 12 = 0 at the x2 = 0, x2 = b, σ 33 = 0, σ 13 = 0, σ 12 = 0 at the x3 = 0, x3 = −a.

(2.33)

Boundary conditions (2.30) − (2.33) show that at each beam face there are four boundary conditions, and then total number of the conditions is equal to 24. In the general case, the equations including the vector of elastic displacements and electrostatic potential have forms of Equations (2.21) – (2.24). Below, we consider the beam oscillations in detail. By taking into account a small cross-section of the beam (i. e. assuming that а/l → 0 and b/l → 0), we can propose in the first approximation that boundary conditions (2.33) carry out in all volume of the sample. By substituting Equations (2.33) into constitutive Equations (2.17) we obtain the next constitutive equations:

ε11 = s11E σ 11 + d 31 E 3 , ε 22 = s12E σ 11 + d 31 E 3 ,

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ε 33 = s13E σ 11 + d 33 E 3 ,

ε12 = 0 , ε 23 = d15 E 2 , ε 13 = d15 E1 ,

(2.34)

D1 = э11T E1 , D2 = э11T E 2 , T D3 = э33 E 3 + d 31σ 11 .

(2.35)

Consider the beam faces without electrodes (i. e. x2 = 0, x2 = b). By taking into account that these faces have normal parallel to the axis Ox2 and boundary conditions (2.30) for small parameter b/l → 0, we can propose in the first approximation that the condition,

D2 = 0

(2.36)

fulfils in all volume of the sample. Then, from Equations (2.34), (2.35) we have

E 2 = 0 , ε 23 = 0 .

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(2.37)

Boundary Problems for Piezoelectrics and Mathematical Models …

23

The beam faces with electrodes (at the x3 = 0, x3 = − a) form the equipotential surfaces stretched in the directions of displacements, and we have in these surfaces

Е1 = Е2 = 0.

(2.38)

Due to the beam depth a is small, then conditions (2.38) carry out in all volume of the sample. Then, we obtain from Equations (2.35), (2.36) that

ε 13 = 0.

(2.39)

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Finally, we obtain two groups of relationships from Equations (2.35), (2.37), (2.39)

ε 12 = ε 13 = ε 23 = 0,

(2.40)

E1 = E 2 = 0, D1 = D2 = 0.

(2.41)

Equations (2.41) have been obtained in Reference [89] by taking into account the small cross-section of the beam and carrying out in the faces with electrodes the conditions Е1 = Е2 = 0. By substituting Equations (2.33) into motion Equations (2.19), we obtain

∂σ 11 ∂ 2 u1 ∂ 2u 2 ∂ 2u3 =ρ 2 , = 0, = 0. ∂x1 ∂t ∂t 2 ∂t 2

(2.42)

Then, represent the displacements as

u1 = u10 e iωt , u2 = u 20 eiωt , u3 = u30 e iωt .

(2.43)

By substituting the last two equations into Equations (2.42), we obtain

u20 = 0, u30 = 0,

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(2.44)

24

Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

u2 = 0, u3 = 0.

(2.45)

Then, by substituting Equations (2.44), (2.45) into Cauchy formulas and taking into account Equations (2.40), we obtain

∂u1 ∂u1 = 0, = 0. ∂x2 ∂x3

(2.46)

Again, by taking into account Equations (2.43), (2.46), we obtain

u1 = u10 ( x1 )e iωt ,

(2.47)

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hence the displacement u1 is a function of two variables, namely x1 and t. Due to u2 = u3 = 0, then ε22 = ε33 = 0, and second and third relationships of Equations (2.34) are not carried out. Therefore, it is necessary to additionally investigate the limit transition from 3-D problem for bar to the beam approximation. From first formula of Equations (2.34), Equations (2.42), (2.47) and first condition of Equations (2.32), we obtain

ε11 = s11E σ 11 + d 31 E 3 , ε 11 =

∂u1 , ∂x1

∂σ 11 ∂ 2 u1 = ρ 2 , u1 = u10 ( x1 ) e iωt , ∂x1 ∂t

(2.48)

and the boundary conditions has form

σ 11 = 0 at the x1 = 0 and x1 = −l.

(2.49)

It should be noted, that the value of E 3 has not still defined in Equations (2.48). By taking into account that below, the low-frequency oscillations will be considered, the electric potential ϕ may be present as

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Boundary Problems for Piezoelectrics and Mathematical Models …

⎞ ⎛ 2 x3 + 1 ⎟ e iω t at the − a ≤ x 3 ≤ 0 , ⎠ ⎝ a ϕ (0) = V0 e iωt , ϕ (− a ) = −V0 e iωt , V0 = const.

25

ϕ ( x 3 ) = −V 0 ⎜

(2.50)

Then we obtain by using last equations

E3 = −

2V0 iωt e . a

(2.51)

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Thus, the electrostatic potential determined by Equations (2.50) corresponds to Equations (2.41) and the conditions for electrodes found by Equation (2.31). Further, by using Equations (2.48), (2.50) state a problem on oscillations of polarized beam with the polarization vector parallel to axis Ox3

∂ 2 u1 1 ∂ 2 u1 , u1 = u10 ( x1 )e iωt , =ρ E 2 2 s11 ∂ x1 ∂t

(2.52)

∂ u1 = d 31V 0 e i ω t at the x1 = 0, x1 = − l . ∂ x1

(2.53)

By substituting u1 into first from Equations (2.52) and into boundary conditions (2.53), we obtain a governing equation and two boundary conditions for calculation of

(v0E ) 2

u10 as

∂ 2 u10 + ω 2 u10 = 0, v E = 2 0 ∂x1

∂u10 = d 31V0 ∂x1

at the

1 , ρ s11E

x1 = 0, x1 = − l .

The solution of Equation (2.54), we shall search in the form

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(2.54)

(2.55)

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Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

u 10 = A sin kx 1 + B cos kx 1 , k =

ω v 0E

.

(2.56)

By using boundary conditions (2.55), we find the constants A and B as

A = d 31

B=

V0 ω , k= E, k v0

(2.57)

d 31V 0 (1 − cos kl ) . k sin kl

(2.58)

Then, by substituting Equations (2.57), (2.58), we obtain the formula for the

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displacement amplitude

u10

as

⎡ ⎛ l ⎞⎤ sin ⎢ k ⎜ x1 + ⎟ ⎥ 2 ⎠⎦ d V ⎣ ⎝ u10 = 31 0 ⋅ , kl k cos 2 where k is the wave number;

k=

ω v

E 0

, v 0E =

1 , ρ s11E

(2.59)

v0E is the phase velocity of propagation of the beam

oscillation mode; ω is the cyclic frequency. Moreover, it is known that

k = 2π λ ,

(2.60)

where λ is the wave length; ρ is the piezoceramic density; V0 is the amplitude of the electric potential at the beam electrodes. Then, the standing wave of the oscillations forms in the beam with the displacement amplitude equal to

u10

u1 = u10 e iωt , Re u1 = u10 cos ω t , Im u1 = u10 sin ω t ,

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(2.61)

Boundary Problems for Piezoelectrics and Mathematical Models …

27

Figure 2.2. Coordinate system (x, y, z) connected with beam.

where

u 10 ( d 31 , V 0 , k , l , x1 ) =

d 31V 0 sin kx , x = x1 + l 2 . k cos( kl / 2 )

(2.62)

Here, the orthogonal coordinate system with centre in the beam middle (see Figure 2.2) is introduced, in which the oscillation amplitude

u10

is the odd

function on the coordinate x . At the point x = 0 , the oscillation amplitude

u10 ( x = 0 ) = 0, i. e. the amplitude in the beam centre is equal to zero, and the oscillation node locates in this point.

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Consider two other proper points

x = ±l 2

or

x1 = 0, x1 = −l .

The

oscillation amplitudes these points are equal numerically and opposite in sign. Equation (2.62) shows that the oscillation amplitude of the beam faces is found as

u10 ( d 31 , V 0 , k , l , 0 ) =

d 31V 0 tg (kl 2 ) . k

(2.63)

Investigate the dependence of the oscillation amplitude of the beam faces on the wave number k. This dependence, constructed by using Equation (2.63), is present in Figure 2.3. Note that the oscillation amplitude of the beam face is equal to infinity at the next values of the wave number:

kl (2 n + 1)π n = 0, 1, 2. = , 2 2

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28

Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

Figure 2.3. Dependence of the oscillation amplitude of the beam faces on the wave number −12 for piezoceramics PZTB-3 ( d 31 = −134 × 10 C / N, vs = 3,340 m/s, V0 = 200 mV ).

Hence, Equation (2.64) describes resonance value of the wave number. Moreover, the oscillation amplitude of the beam faces is equal to zero at the wave number

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kl / 2 = nπ .

(2.65)

By assuming that n = 0 in Equation (2.65) and using the definition of the wave number (2.60), we calculate first resonance frequency

f R(1) of the mode

BLt of longitudinal oscillations perpendicular to the polarization vector as

f Rl =

v 0E , ω R = 2 π f Rl , 2l

(2.66)

where ωR is the corresponding cyclic frequency. The total complex conductivity of the beam could be obtained by using ratio of displacement current to applied voltage. The displacement current could be determined by using Equations (2.35), (2.59), (2.63) as l

2 kl ⎤ ⎡ T J c = iω b ∫ D3 dx1 = iω bl э33 E 3 ⎢1 − k 312 + k 312 tg ⎥ . kl 2⎦ ⎣ 0

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(2.67)

Boundary Problems for Piezoelectrics and Mathematical Models …

29

The electric potential and electric field intensity are coupled by the dependence

E3 = −

∂ϕ ∂x 3

.

Due to the value of

E3

is the constant, then we obtain

E3 a = ϕ (0) − ϕ (a) = ΔV .

(2.68)

Then the total conductivity of the beam on the transverse mode BLt may be obtained as T lbэ33 1 = Y1 (ω ) = iω a Z1

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where χ =

ω v 0E

⎛ 2 χ⎞ ⎜⎜1 − k 312 + k 312 tg ⎟⎟ , χ 2⎠ ⎝

(2.69)

l = kl .

Equation (2.69) shows that in absence of losses, the beam conductivity has only reactive component. The frequencies of resonance and antiresonance coincide respectively with pole and zero of the function Y1 (ω ) . The conductivity of the beam without loading is equal to infinity at the resonance frequency in absence of losses. Then, by substituting the value of Y1 → ∞ (the resonance condition) into Equation (2.69), we obtain χ R

=π .

By using the frequency of dynamical resonance the relationship

χ R = ω R l ρ s11E

elastic compliance factor

s11E =

f Rl and taking into account

, it may be obtained an equation for the

s11E in the form

1 . 4 ρ l ( f Rl ) 2 2

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30

Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al. Due to the total conductivity is equal to Y1 (ω ) = 0 on the antiresonance

frequency, then, by using Equation (2.69), we obtain the relationship for the electromechanical couple factor

k 312 =

1 1−

2

χA

tg

χA

k 31 as

.

(2.71)

2

By substituting in this equation the formula χ A =

π f Al f

l R

l

, where f A is the

linear frequency of first antiresonance of the oscillation mode BLt, then we finally obtain

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k 312 =

1 2 f Rl ⎛ π f Al ⎞ ⎟ 1− tg ⎜ π f Al ⎜⎝ 2 f Rl ⎟⎠

.

(2.72)

2.3. PROBLEM ON LONGITUDINAL OSCILLATIONS OF BEAM UNDER LONGITUDINAL ELECTRIC FIELD This problem only differs from the considered one in Paragraph 2.2 by the electric boundary conditions. Consider the problem on the oscillations of preliminary polarized beam of rectangular cross-section with volume V = {0 ≤ x1 ≤ h, 0 ≤ x 2 ≤ w, 0 ≤ x3 ≤ a} in the orthogonal system of coordinates Ox1x2x3. The direction of the polarization field coincides with the positive direction of axis Ox3 (see Figure 2.4). The electrodes are placed in the face surfaces x3 = 0 and x3 = a. The length of the beam significantly exceeds its transverse sizes.

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Figure 2.4. Scheme of longitudinal oscillations of the beam.

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In the case when the dielectric permeability of the beam significantly exceeds the dielectric permeability of air (the scattering electric fields are absent) and electric force lines are parallel to the beam length, then the electric boundary conditions at the side faces (by taking into account the electrode location) may be present as

D1 = D 2 = 0 .

(2.73)

By taking into account the small cross-section of the beam in compare with its length, it could be assumed that conditions (2.73) are fulfilled in all volume of the sample. Then, by substituting Equations (2.73) into Maxwell electrostatic equation div D = 0, we obtain

∂D 3 =0. ∂x 3

(2.74)

Due to the faces with electrodes (x3 = 0 and x3 = a) forms equipotential surfaces, then it could be assumed that in them, the electric field fulfils the conditions as

E1 = 0, E 2 = 0 .

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32

Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al. Now consider mechanical boundary conditions. In the case of low-frequency

oscillation modes, an elementary volume is assumed free (i. e. transverse direction. Therefore, we select the stresses

σ kl

σ kl = 0 ) in the as independent

variables. Due to the transverse sizes of the beam are small in compare with its length (there is a 1-D oscillation mode), then the boundary mechanical conditions carry out in all volume of the sample. The side surfaces of the beam are free from mechanical stresses, and the next boundary conditions are fulfilled as

σ 11 = 0, σ 22 = 0 at the 0 ≤ x1 ≤ h, 0 ≤ x 2 ≤ w, σ 12 = 0, σ 23 = 0, σ 13 = 0.

(2.76)

The constitutive equations of piezoceramics by taking into account as the independent variables of the mechanical stresses and electric induction are stated as [89] D ε ij = s ijkl σ kl − g kij D k ,

(2.77)

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E i = g ikl σ kl + β ikT D k . Rewrite these equations in the matrix designations as

S i = sijD T j + g ij D j ,

(2.78)

E i = − g ij T j + β ijT D j , where

⎛ ∂E g ij = ⎜ − i ⎜ ∂T j ⎝

⎞ ⎟ are the pressure piezoconstants characterizing the ⎟ ⎠

intensity of electric field of the idle motion in piezoelement at the given mechanical stress;

⎛ ∂E ⎞

i ⎟⎟ β kiT = ⎜⎜ ∂ D k ⎠T ⎝

are the dielectric permeability at the

stresses T = 0 or T = const. By taking into account Equations (2.73) – (2.76), we obtain the constitutive equations for the longitudinal oscillations in the form

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ε 33 = s 33D σ 33 + g 33 D3 ,

33

(2.79)

E 3 = − g 33σ 33 + β 33T D3 .

Then, we state the motion equations for the problem considered by substituting Equations (2.76) into motion Equation (2.19) as

∂ σ 33 ∂ 2 u 3 ∂ 2 u1 ∂ 2u 2 =ρ , = 0, = 0. ∂x 3 ∂t 2 ∂t 2 ∂t 2 Present the displacements

u1 , u 2 , u3

(2.80)

in the form of Equations (2.43). By

substituting two first relationships from Equations (2.43) into Equations (2.80), we obtain

u10 = 0, u 20 = 0,

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u1 = 0, u 2 = 0.

(2.81)

Then, by substituting Equations (2.81) into the Cauchy relationships and taking into account above assumptions, we obtain

∂u 3 ∂u 3 = 0, = 0. ∂x1 ∂x 2

(2.82)

Further, Equations (2.43) and (2.82) lead to equation

u3 = u30 ( x3 ) e

iω t

.

(2.83)

This representation signs that the displacement u 3 is the function of two variables, namely x3 and t. Thus, by using first Equations (2.79), (2.80), conditions (2.73), (2.76), and Equation (2.83), we obtain

ε 33 = s33D σ 33 − g 33 D3 , ε 33 =

∂u 3 , ∂x 3

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34

Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

∂σ 33 ∂ 2u = ρ 23 , u3 = u30 ( x3 ) e iω t , ∂x 3 ∂t σ 33 = 0 at the x 3 = 0 , x 3 = a , E1 = 0, E 2 = 0, E 3 = V 0 e

iω t

(2.85)

(2.86)

.

The governing equation for the problem on forced oscillations of the beam is stated by using Cauchy relationship

ε 33 = du 3 / dx3

and Equations (2.84) −

(2.86) as

d 2u3 iω t + ρω 2 s33D u 3 = 0, u 3 = u 30 ( x3 )e . 2 dx3

(2.87)

This equation, it should be added the boundary conditions:

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du3 = g 33 D3 at the x3 = 0, x3 = a. dx3

(2.88)

The solution of the boundary problem (2.87) – (2.88) is defined as

u 3 ( x3 ) = where

g 33 D 3 χ (sin kx 3 − tg cos kx 3 ), k 2

(2.89)

D k 2 = ρω 2 s33 , and χ = ka is the dimensionless frequency.

Due to definition of E 3 = −∂ ϕ / ∂x 3 the voltage between the face electrodes is obtained in the form a

ΔV = −∫ Ez dz .

(2.90)

0

By carrying out the integration of last relationship from Equations (2.79), we determine by using Equation (2.90) the next relationship:

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ΔV = −

2 ⎛ g 33 g 33 T ⎞ ⎟ aD 3 . ⎜ [ u ( a ) u ( 0 )] β − + + 3 3 33 D D ⎟ ⎜s s 33 ⎠ ⎝ 33

35

(2.91)

Then, by substituting Equation (2.89) into Equation (2.91), we obtain

⎛ 2 χ⎞ ΔV = aβ 33T D 3 ⎜⎜1 + k D2 − k D2 tg ⎟⎟, χ 2⎠ ⎝ where the dynamical factor of electromechanical couple k D2 =

(2.92)

2 g 33 D s 33

β

T 33

is

determined by using well-known definition of static factor of electromechanical couple [124]:

k

2 33

We k D2 = = . Wm + We 1 + k D2

(2.93)

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where We, Wm is the electric and mechanical power, respectively. Then, by division of Equation (2.92) on the displacement current I d = i ω hwD 3 , it is

stated a relationship for the beam impedance into longitudinal field:

Z2 =

aβ 33T ⎛ 2 χ⎞ ⎜⎜ 1 + k D2 − k D2 tg ⎟⎟. χ 2⎠ iω hw ⎝

(2.94)

By neglecting losses, the beam impedance has purely reactive character. In this case, the frequencies of resonance fR and antiresonance fA coincide with zeroes and poles of the function Z2(ω), respectively. If we assume that Z2 = 0 into Equation (2.94) and take into account relationship (2.93), then the equation for the factor of electromechanical couple is present as

k 332 =

χR 2

ctg

χR 2

.

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Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

By using the condition of the antiresonace Z 2 → ∞ (at the χ A = π ) , we obtain from Equation (2.95) the final equations for the factors of elastic compliance as

s 33D 1 π2 E , = , s 33 = s = 1 − k 332 ρω A2 a 2 4 ρ ( f At ) 2 a 2 D 33

(2.96)

t

where ωA is the cyclic frequency of antiresonance and f A is the thickness frequency of antiresonance.

2.4. PROBLEM ON SHEAR OSCILLATIONS EXCITED IN PLATE WITH DEPTH POLARIZATION Consider the oscillations on shear mode of plate with depth polarization and having volume V = {0 ≤ x1 ≤ l 2 , 0 ≤ x 2 ≤ w, 0 ≤ x3 ≤ b} in the orthogonal

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system of coordinates Ox1x2x3. The direction of the polarization field coincides with the positive direction of axis Ox3 (see Figure 2.5).

Figure 2.5. Scheme of shear oscillations of the piezoelement.

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37

The electrodes are placed in the side faces of the plate (at the x1 = 0 and x1 = l2). The excited mode is connected with wave propagation along axis Ox1 of displacements being parallel to axis Ox3. The electric field with voltage ΔV is applied in perpendicular direction to the polarization vector. The surfaces with electrodes are equipotential, hence in them (at the x1 = 0, l 2 ) the electric

potential is a constant ( ϕ = const ). In this case, the planes parallel to them will be also equipotential, and therefore the electric field is homogeneous in plane, i. e.

E1 = const. Hence, the components of the electric induction vector D2 , D3 are equal zero. Therefore, the electrostatic equations lead to the equation

∂ D1 / ∂ x1 = 0 or D1 = const. Due to the piezoelectric oscillations are excited by the electric field of intensity E1 , then by taking into account equality ϕ = const , it could be assumed at the faces x1 = 0 and x1 = l2 that

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E 2 = 0, E 3 = 0,

dE 1 = 0. dx 1

(2.97)

Now consider mechanical boundary conditions. The coupled electromechanical state of the plate (Figure 2.5) from transverse-isotropic material will be only characterized by the shear strain ε 13 and tangential stress σ 13 . All of other stresses are piezoelectrically inactive. Therefore, the next mechanical boundary conditions are fulfilled in the faces without electrodes as

σ 11 = 0, σ 22 = 0, σ 33 = 0, σ 12 = 0, σ 23 = 0 .

(2.98)

Due to the sizes in the wide plane of the plate are greater then its thickness, the next condition is carried out: l2 ≥ w ≥ b . Hence, it could be assumed that Equations (2.97), (2.98) are fulfilled into all volume of the plate. By taking into account these conclusions, as the independent variables may be considered deformations and electric induction. Then constitutive Equations (2.7) are selected as

σ 13 = c 55D ε 13 − h51 D1 , E1 = − h15 ε 13 + β 11S D1 .

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Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

The governing equation for the problem on forced shear oscillations of the plate is obtained by using Cauchy relationship ε 13 = ∂u 3 / ∂x1 and Equations (2.97) – (2.99) as

d 2u3 iω t + ρω 2 s 55D u 3 = 0, u 3 = u 30 ( x1 ) e . 2 dx 1

(2.100)

This equation should be added by the boundary conditions:

du 3 h15 = D D1 at the x1 = 0 , x1 = l 2 . dx 1 c 55

(2.101)

Due to Equations (2.100), (2.101) coincide with Equations (2.87), (2.88) at the replacements x1 → x3,

D D = 1 / c 55D → s33 h15 / c55D → g 33 and s 55 , then the

solution of boundary problems (2.100), (2.101) has form which is analogous to Equation (2.89):

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u 3 ( x1 ) =

where

h15 D 1 χ (sin kx 1 − tg cos kx 1 ) , D 2 c 55 k

(2.102)

k = ρω 2 / c 55D , and χ = kb is the dimensionless frequency.

The voltage between electrodes may be presents trough D1. By integrating the equation on the variable of x1 and then dividing on the displacement current

J d = iω l 2 bD1 , we obtain a relationship for the plate impedance on shear mode

as

l 2 β 11S Z5 = i ω bw where

χ = ω l2

⎛ χ⎞ 2 ⎜⎜ 1 − k 152 tg ⎟⎟ , χ 2⎠ ⎝

ρ .

D c 55

(2.103)

(2.104)

Equation (2.102) shows that the dielectric permeability of the free sample in the direction which is perpendicular to the polarization vector could be measured

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39

through the capacitance on low frequency. By equating the relationship for Z 5 to zero in correspondence with the resonance condition in absence of losses in the sample, we obtain the shear factor of electromechanical couple as

k

where

2 15

=

χR 2

ctg

χR 2

=

π f Rs 2 f As

⎛ π f Rs ctg ⎜⎜ s ⎝ 2 fA

⎞ ⎟⎟ , ⎠

(2.105)

f Rs , f As are the frequencies of resonance and antiresonance at the shear

oscillations. Then, by taking into account the cyclic frequency of resonance ω R

= 2πf Rs ,

and that χ R = π , we obtain by using relation (2.104) the shear module as

c 55D =

ω R2 l 22 ρ = 4 ( f Rs ) 2 l 22 ρ 2 χR

.

(2.106)

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Due to symmetry conditions for transtropic materials (2.15), we finally obtain the relationship: D D . c 44 = c55

(2.107)

2.5. PROBLEM ON PLANAR OSCILLATIONS IN THIN PLATE WITH DEPTH POLARIZATION. APPROXIMATE RELATIONS FOR ELASTIC CONSTANTS The planar forms of oscillations have been investigated by R. Bechman, A. V. Belocon, V. T. Grinchenko, V. L. Karlash, M. A. Meduck, V. V. Meleshko, Y. H. Pao, A. F. Ulitko, et al. [29, 37, 62, 91, 92, 101]. Consider the problem on the steady planar oscillations of rectangular thin plate with volume V = {−a ≤ x1 ≤ a, − b ≤ x 2 ≤ b, − h ≤ x3 ≤ h} in the orthogonal system of coordinates Ox1x2x3 (see Figure 2.6). The plate is polarized and the direction of the polarization vector Ps coincides with the positive direction

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Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

of axis Ox3. The excitation of the oscillation is carried out by voltage 2V0 e

iω t

harmonically changing in the time. This voltage is applied to the electrodes totally covering both surfaces of the plate (at the x3 = ± h).

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Figure 2.6. Scheme of planar oscillations of the square plate.

The constitutive equations of the planar oscillations of rectangular thin plates could be stated in simplification of 3-D dynamical equations for piezoelectric medium [61, 62, 89] by using the hypotheses of the generalized plane stress state. It has been shown [61, 62] that the problem on steady oscillations of piezoceramic plate could be reduced to the ‘purely’ mechanical problem on the forced oscillations of the rectangular thin plate loaded by uniformly-distributed normal stresses applied to its side surfaces. These stresses are determined as

m V0 iω t m V0 iω t 1 1 σ 22 = −d 31 e , (2.108) σ 11 = − d 31 e , 2G m−2 h m−2 h 2G where

s12E m = 1+ E , σ = − E , σ s11 1

E

and G is the shear module.

The above transformation of the initial problem is carried out by using an averaging on the plate depth, and then, it is solved by using mean values of the displacement amplitudes

u1 ( x1 , x2 ) and u 2 ( x1 , x 2 ) .

The boundary problem by taking into account Equation system (1.1) from Reference [61] has allowed one to obtain an exact analytic solution in

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41

trigonometric rows [60]. The analysis this cumbersome solution has shown that electric potential ϕ may be very well approximated as

ϕ=

V0 iω t ze . h

(2.109)

This equation is not only applied into narrow ranges of frequencies including own frequencies of mechanical oscillations of the plate. The problem on the forced oscillations of the rectangular plate from isotropic elastic material (with the Poisson’s ratio of σ = 0.333 ) has been numerically solved [62] for six discrete values of resonance frequencies. The calculations have shown an increase of the result error for solution of the boundary problem near own frequency of oscillations. Above method [62] has also been applied to investigation of planar oscillations of the rectangular piezoceramic plates [61]. In this case, there has been calculated the plate oscillations for nine first values of normalized resonance frequencies

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Ω=

ωb vD

where v D =

,

2

(2.110)

m −1 G is the velocity of the expansion waves. m−2 ρ

The values of resonance frequencies for one of the boundary oscillation modes (of expansion) for rectangular plates for several values of the plate geometry have been also found by the test method of ‘resonance-antiresonance’ [61]. The theoretical analysis four different types of modes, including the contour oscillation modes for thin square plates, has been fulfilled in Reference [29]. By using the constitutive equations for piezoceramics (2.17), in which the mechanical stresses and intensity of electric field to be independent variables, there have been obtained the governing equations for mean values of the mechanical stresses on the plate thickness as

σ 11 = γ 11ε 11 + γ 12 ε 22 + γ 16 ε 12 ,

(2.111)

σ 22 = γ 12 ε 11 + γ 22 ε 22 + γ 26 ε 12 ,

(2.112)

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Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al.

σ 12 = γ 13ε 11 + γ 23ε 22 + γ 66 ε 12 , where

γ ik = γ ki

(2.113)

are the stiffness constants defined for various oscillation

modes. In the case of the plate from quasiisotropic material, there are next relationships:

s11 = s 22 , s16 = 0, s 26 = 0. In Reference [29], there have been stated relationships for

(2.114)

γ ik

in the form

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⎧ s11 E = , ⎪γ 11 = γ 22 = 2 2 s11 − s12 1 − σ 2 ⎪ ⎪ s12 Eσ = , ⎨γ 12 = − 2 2 2 − 1 − σ s s 11 12 ⎪ ⎪ 1 , γ 16 = γ 26 = 0, ⎪γ 66 = s66 ⎩ where

E=

(2.115)

s γ 1 is Young module, σ = − 12 = 12 is Poisson ratio. s11 γ 11 s11

In the case of the more general conditions, when there are relationships

s11 = s 22 , s16 = s 26 , the representations for

γ ik

(2.116)

have forms

⎧Πγ 11 = Πγ 22 = s11 s66 − s162 , ⎪ 2 ⎪Πγ 12 = s16 − s12 s66 , ⎪ 2 2 ⎨Πγ 66 = s11 − s12 , ⎪Πγ = Πγ = ( s − s ) s , 26 12 11 16 ⎪ 16 ⎪Π = ( s112 − s122 ) s66 − 2( s11 − s12 ) s162 . ⎩

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Boundary Problems for Piezoelectrics and Mathematical Models …

43

If there are conditions (2.114) and (2.116), then it is carried out the equation,

( s11 − s12 )(γ 11 − γ 12 ) = 1.

Above-mentioned equations for

γ ik

(i, k = 1, 2, 6)

are only applied to the plane perpendicular to axis Ox3. Here, we remove for brevity upper index E at the compliance factors

sij . This index signs that these

factors are measured into constant electric field E = const . The frequency of own oscillations of the plate f = ω / 2π (where ω is the cyclic frequency) may be calculated for different modes from the well-known relationship [89]:

ρω 2 k2

=γ

,

(2.118)

ρ

is the material density, k = π / a is the frequency constant, γ is the stiffness constant determined by the relationships for proper oscillation modes. These relationships for four oscillation modes of square plate have been obtained in Reference [29] where

γ = γ 11 − γ 12 ,

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for longitudinal Mode I as for longitudinal Mode II as

γ = γ 11 ,

for longitudinal Mode III as

γ = γ 11 +

for contour shear mode as

(2.119) (2.120)

8

π2

γ 12 ,

γ = γ 66 .

(2.121)

(2.122)

It has been experimentally stated that, the resonances another modes and their harmonics lay upon resonances these oscillation modes. Therefore, it is very difficult to select the resonance separated from other modes. This procedure may be reliably carried out only for longitudinal Mode III observed at the fulfillment of Equations (2.114). Hence, the resonance frequency of Mode III could be stated with higher accuracy in compare with the frequencies other oscillation modes.

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State an equation for calculation of resonance frequency of longitudinal contour Mode III. With this aim, we substitute relationship (2.121) into Equation (2.118), and by taking into account that

ω c = 2πf c , obtain

π 2 s11 − 8 s12 , 4π 2 a 2 ρ (s112 − s122 )

f IIIc =

(2.123)

where а is the side size of the square plate, the resonance frequency of plate c

oscillations f III connected with the resonance cyclic frequency relationship

ω c through

f IIIc = ω c / 2π . Moreover, Equation (2.123) determines the E

compliance factor s12 as

s = E 12

2 ± 16 − a 2π 4 ρ ( f IIIc ) 2 [ s11E − 4 a 2 ρ ( f IIIc ) 2 ( s11E ) 2 ] 2π 2 ρ ( f IIIc ) 2 c

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Thus, by measuring the resonance frequency f III and knowing E

. (2.124)

E , it may s11 E

be calculated the constant s12 . However, a calculation error of the constant s11 is added to calculation of

s12E

by using Equation (2.124). Therefore, it is E

preliminary necessary to estimate the error of the constant s11 . This procedure is E

fulfilled by using a calculation of s12 from other alternative method. That method to identify various counter modes of thin square plates and calculation their elastic constants has been proposed in Reference [30] after discovery additional fourth counter longitudinal mode of the plate oscillations named the Baerwald’s mode (this is Mode IV) [24]. The method differing from previous one has been developed on the base of more total identification of various counter modes for square plates. This method is present below. An analysis of various oscillation modes in rectangular plate, based on the representation of the displacement components by using Legendre polynomials

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45

⎛x ⎞ Pn ⎜ 1 ⎟ , has shown that the forces in the plate surface F j(n ) could be calculated ⎝a⎠ by using the stresses σ 1 j at the edge of the plate [91, 101] as

⎡ ⎛x ⎞ ⎤ F j(n ) = ⎢ Pn ⎜ 1 ⎟σ 1 j ⎥ . ⎣ ⎝ a ⎠ ⎦ x =± a

(2.125)

1

Moreover, it has been shown [101] that the frequency constants c N III = f IIIc d and N IV = f IV d could be calculated by using experimental

values of resonance frequencies of three oscillation modes of circular thin disc with diameter d. The values of FIII and FIV have been calculated by using the table data of the dependencies FIII (σ ) and FIV (σ ) from Reference [101]. These E

values allowed one to define the corresponding compliance factors s11 as

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s11E

Fd2 , = 4ρ N d

(2.126)

where Fd = FIII , N d = N III or Fd = FIV , N d = N IV , respectively. Then, it has been experimentally proved that as for circular discs as for thin c

c

square plates, the ratio of the corresponding oscillation frequencies f III and f IV satisfies to the next relationship:

f IVc F (σ ) = IV . c FIII (σ ) f III

(2.127)

By using this equation, we developed the calculation method for the E

compliance factor s12 , based on test measurements of the resonance frequencies

f IIIc and f IVc for counter Mode III and Mode IV of the thin square plate oscillations.

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The core of the test method consists of that, we substitute into Equation (2.126) instead of Fd and N d , the values of FIII , N III and FIV , N IV obtained from measurements of the resonance frequencies of the longitudinal Mode III and the Baerwald’s Mode IV of the square plate. Then, it may be used Equation c

c

(2.127) for the ratio of the resonance frequencies f III and f IV . Equation (2.127) allows one to find the value of FIV / FIII on the base of the measured frequencies c

c

ratio f III / f IV . Further, by using the normalized values of the dependence of the ratio FIV (σ ) / FIII (σ ) , presented into Table IV of Reference [24] for the calculated value of FIV / FIII , we find the magnitude of the Poisson’s ratio Previously calculating the value of the compliance factor

σ E.

s11E by using

Equation (2.70) for the resonance frequency of long beam f Rl and substituting the result obtained into Equation (2.124), it may be stated a value of compliance E

factor s12 as

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s12E = −σ E s11E . A comparison this value of

(2.128)

s12E

with its value calculated from Equation

(2.127) allows one to estimate the error of determination this constant. Further, we obtain a relationship for the planar factor of electro-mechanic coupling k p corresponding to the polarization plane perpendicular to axis Ox3. This factor will be used below. It is known that by neglecting heat and magnetic terms, the internal energy of linear electroelastic system is determined as [89]

U =

1 1 ε i Ti + D m E m , i = 1, 2, ..., 6, m = 1, 2, ..., 3 . 2 2

(2.129)

This relationship may be rewritten by using constitutive Equation (2.13) in the form

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47

1 1 1 1 T U = Ti sijET j + Ti d mi Em + Em d miTi + Em эmk Ek = Ue + 2U m + U d . 2 2 2 2 (2.130) Then the factor of electromechanical couple may be present through the interaction energy ( U m ), elastic energy ( U e ) and electric energy ( U d ) as

k=

Um U eU d

.

Hence, Equations (2.129), (2.130) in the case of electric field parallel to the axis Ox3 lead to final relationship:

kp =

d 31 T э 33 s11E

2 1−σ

E

= k 31

2 1−σ

E

.

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Thus, we can obtain the planar factor of electromechanical couple calculation of

k 31 by using Equation (2.71).

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(2.131)

kp

after

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Chapter 3

METHODS FOR DEFINITION OF PIEZOELECTRIC MATERIAL CONSTANTS

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3.1. METHODS FOR DEFINITION OF ELASTIC PIEZOELECTRIC CONSTANTS IN DYNAMIC REGIME There are many different methods for definition of elastic constants of the construction materials which may be divided into two large groups related to isotropic and anisotropic materials [1, 2, 26, 44, 64, 69, 71, 84, 99, 102, 103, 116, 130, 139, 141, 143]. The systematization these methods could be carried out on the base of various criteria, for example (i) form and character of oscillations, (ii) method their excitation and registration, (iii) thermal range, (iv) scaling of oscillation amplitude, (v) oscillation frequency, (vi) type of sample deformation, etc. The different classifications based on particular criteria have been presented in References [1, 57, 106, 108]. One of the most total classifications of research methods of elastic and deformation material properties almost adequate to the methods of definition of the elastic material constants is the classification suggested in Reference [1]. Its constitutive parameter is the loading frequency. In according with this classification, all methods of investigation of the properties or determination of the elastic constants are divided into several groups, namely (i) wave methods, (ii) heat methods, (iii) resonance methods, (iv) methods of oscillation damping, (v) phase methods, (vi) hysteresis methods, (vii) pendulum methods, and (viii) quasistatic methods. This classification is convenient due to the each method considered, it is corresponded a frequency range into which the method could be

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used. Due to this circumstance, the application of that classification allows one to easily define the loading frequencies uncovering by the known methods. Based on the loading frequency, all methods of definition of the piezoactive material constants could be divided into two groups, namely the quasistatic (at the loading frequencies of Hertz parts) and dynamic (at the loading frequencies from some tens of Hertz up to Mega Hertz) ones (see Figure 3.1). The quasistatic methods could be united in groups on the base of the measuring method of sample deformation. Most wide-spread method of this group is the known tensiometric one. This method has been used in investigation of piezoelectric materials [98]. There have been obtained values of Young module and Poisson’s ratio at static compression and tension of some piezoceramic samples. The measuring method of deformations by using movable-electrode tubes differ from above-mentioned by application to measurement of strains the special electro-vacuum lamps − the movable-electrode tubes. These quasistatic methods have one general property, namely the sample loading is carried out by using mechanical loading (tension, compression, bending, torsion, shear and also their combinations). The mechanical loading method differing from above test methods by several features is also applied in two known methods of definition of the shear module [28]. The core of first method is the loading of a sample in the form of ring first by two and then by three concentrated forces, the measurement of deformations and calculation of the shear by using the strain values. Second method differs by application of three-point bending of the beams with hinge base and with consolpinched edges. All enumerated quasistatic methods of definition of the elastic constants have one general lack, namely by using these methods may be defined no more than four constants of piezomaterial. Moreover, the last method [56] may be used only for the piezoceramics with sufficiently high Curie temperature because at the relatively low temperatures (∼ 60 − 80 °C) of heat that not exceed the phase transition temperature, the thermal elongation of the sample is small, and it could not be measured with sufficiently high accuracy in general case. Due to the estimation of total set of the elastic constants by using above methods is difficult, then their value is low, and the applications are limited. The dynamic measuring methods of elastic constants differ from static ones by higher frequencies of loading. Their diapason begins from some Hertz for the methods of bending and torsion pendulums and increase up to 106–107 Hz for the method of ‘resonance-antiresonance’ (MRA) and the impulse method.

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Methods for Definition of Piezoelectric Material Constants

51

Test methods for definition of constants of piezoactive materials

Dynamic methods

Method of ‘resonance − antiresonance’

Impulse method

Static and quasistatic methods

Tensiometric method

Method of resonance spectra

Method of mechanical tensiometers

Method for definition of total set of constants on samples of one form

Method of optical tensiometers

Method of Q-meter

Method of measurement of strains by using movable-electrode tubes

Method of electric charge

Sclerometric methods

Pendulum methods

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Figure 3.1. Classification of methods for definition of constants of the piezoactive materials.

Some dynamic methods allow one to define a set of piezoceramic constants with sufficiently high accuracy. In particular, among them, the method of resonance spectra (MRS) [108] by using of which it may be calculated a set of elastic resonances of piezomaterials in ferro- or paraphase in thermal treatment including temperatures near the temperature of ferroelectric phase transition. In the MRS, it is used spontaneous (or induced) unipolarity of piezoceramics. This method allows one to measure elastic modules and Poisson’s ratio with the error of 0.8 − 1 % and 0.03 %, respectively that is smaller than the errors of MRA. However, by using MRS, it is not possible to define total set of piezoceramic constants that decreases a range its applications. Among other dynamic methods based on excitation of oscillation of 20 kHz and more in a sample, it is also known the method of electromechanical Q-meter [2, 123]. In this method the elastic constants are found on the base of resonance frequencies of the composite bar consisting of aluminum bars with bars of test material glued on their faces. The method of Q-meter allows one to measure only three constants of piezoceramics with the errors of 3 − 5 %, and therefore now this method is seldom used. Most wide-spread test method for ferroelectrics and piezoactive materials is the MRA, regulated by different internal standards [64, 69, 116] and allowing one

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to determine total set of piezoceramic constants. The core of the method of ‘resonance-antiresonance’ consists in the next. From material of investigated specimen processed by using ceramic or another technique [64, 99], the samples of three geometric forms are prepared, namely (i) the circular thin plate with t 0 . The

are caused by piezoeffect and define, respectively

piezoelectric and dielectric properties, moreover negatively-definite matrix,

165

K ϕϕ ≥ 0 1.

K ϕϕ

The vectors

is symmetric and no

Fu

and

Fϕ

form in a

result of account of the mechanical and electric influences. The finite-element objects caused by acoustic medium could be present as

~ R *uψ = ε w R *uψ , R *uψ = ρ w

∫ Nψ ⋅ (n

w

⋅ N *u ) dS,

(A.20)

S hws

Mψψ

ρw

2 ρ w2ε w = 2 ∫ Nψ ⋅ Nψ dΩ + cw Ωhw Z *

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Cψψ =εwKψψ +

ρw2 Z

∫ Nψ ⋅ Nψ d S , *

(A.21)

Shwi

∫Nψ ⋅ Nψ dS, Kψψ = ρ ∫Bψ ⋅Bψ dΩ, Bψ =∇Nψ , *

*

w

Shwi

Ωhw

(A.22)

S hw = ∂ Ω w , S hw = S hwf U S hwc U S hwi U S hws , εw =

b 2 ρ wcw

.

(A.23)

Here * is the conjugation operation. The characteristic feature of form of Equations (A.17) − (A.19) in ACELAN is the symmetry of the matrices M and K, and also their one-type saddle structure. The last designs that the matrices M and K after some simultaneous rearrangement their rows and columns reduce to the next block forms:

1

Here the inequalities B > 0 or B ≥ 0 design that the matrix B is respectively positively-definite or semi-definite one.

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⎛ B A = ⎜⎜ T ⎝H

H⎞ ⎟⎟ ; B = B T , S = S T , B ≥ 0 , S ≥ 0 , − S⎠

(A.24)

and the sizes of all blocks for the matrices M and K, and also for the matrix C are the same. In the case of steady oscillations, after substitution

~ ~ Fu = Fu ( x ) exp( iω t ) , Fϕ = Fϕ ( x ) exp( i ω t ) ,

a = ~ a ( x ) exp( i ω t ) ,

(A.25)

we obtain the next linear system of algebraic equations (LSAE):

~ ~ ~ ~ T Kc ⋅ ~ a = Fc , Fc = [ Fu , Fϕ ,0] ,

(A.26)

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where

⎛ K uuc ⎜ K c = ⎜ K Tuϕ ⎜ KT ⎝ uψ c

K uϕ − K ϕϕc 0

K uψc ⎞ ⎟ 0 ⎟ − K ψψc ⎟⎠

,

(A.27)

Kηηc = −ω 2Mηη + iω Cηη + Kηη , η = u , ψ ,

(A.28)

~ K uψc = −ω 2 R uψ + iω R uψ , K ϕϕ c =

(A.29)

1 K ϕϕ . (1 + i ωζ d )

If use a real arithmetic, then problem (A.26) − (A.29) presents LSAE relatively of vector of the real and imagine parts of the node degrees of freedom of a with matrix not demonstrating the positive definiteness and saddle structure. In order to solve LSAE (A.26) − (A.29), the Peidge-Saunders iteration method for non-symmetric matrices is used in ACELAN.

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167

A.1.3. Numerical Solution of System of the Ordinary Differential Equations for the Finite-Element Model In the case of non-stationary problems, it is used Newmark’s method in alternative formulation [93]. With this aim, apply to Equation (A.17) in ∀ j-th time layer tj = jτ (τ = Δt is the time step) the averaging operators Yj in according with the formulas: 2

Y j b = ∑ β k b j +1−k , b = a, Fu , Fϕ ,

(A.30)

1 Y j a& = [γ a j +1 − (2γ − 1)a j − (1 − γ )a j −1 ] ,

(A.31)

k =0

τ

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Y j &a& =

1

τ2

(a j +1 − 2a j +a j −1 ) ,

(A.32)

where b j = b(t j ) , β 0 = β , β1 = γ 1 − 2 β , β 2 = γ 2 + β , γ 1 = 1 / 2 + γ , γ 2 = 1/ 2 − γ . Here β, γ are the parameters of the Newmark’s method which is undoubtedly stable at the 4 β ≥ (1 / 2 + γ ) 2 , γ ≥ 1/2. As a result, we obtain from Equations (A.17) – (A.19), (A.26) – (A.32) the implicit scheme with step on tj in the form

K eff ⋅ a j +1 = F eff (Y j Fu , Y j Fϕ , a j , a j −1 ) ,

(A.33)

where

K eff

eff ⎛ K uu ⎜ = ⎜ K Tuϕ ⎜ K eff T ⎝ uψ

K uϕ eff − K ϕϕ 0

K ueffψ ⎞ ⎟ 0 ⎟, eff ⎟ − Kψψ ⎠

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(A.34)

168

Vladimir A. Akopyan, Arkady N. Soloviev, Ivan A. Parinov et al. eff Kηη =

1

βτ

2

Mηη +

γ C + Kηη , η = u , ψ , βτ ηη −1

⎛ ζ dγ ⎞ γ 1 ~ eff K ⎟ K , ⎜⎜1 + = K = R + R , ϕϕ βτ ⎟⎠ ϕϕ βτ 2 uψ βτ uψ ⎝ 1 1 1 ~ Fueff, j+1 = Yj Fu + 2 Muu ⋅ (2U j −U j−1) + 2 Ruψ ⋅ (2Ψ j − Ψ j−1) + β βτ βτ 1 1 + Cuu ⋅[(2γ −1)U j + (1−γ )U j−1] + Ruψ ⋅[(2γ −1)Ψ j + (1− γ )Ψ j−1 ] − βτ βτ eff uψ

β β ⎛β ⎞ ⎛β ⎞ − Kuu ⋅ ⎜⎜ 1 U j + 2 U j−1 ⎟⎟ − Kuϕ ⋅ ⎜⎜ 1 Φj + 2 Φj−1 ⎟⎟, β β ⎝β ⎠ ⎝β ⎠ ζ ζ 1 Fϕeff, j +1 = Yj Fϕ + d Yj F&ϕ + d KTuϕ ⋅[(2γ −1)U j + (1 − γ )U j −1 ] −

β

βτ

βτ

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⎛β ⎞ ⎛β ⎞ β β − KTuϕ ⋅ ⎜⎜ 1 U j + 2 U j −1 ⎟⎟ + Kϕϕ ⋅ ⎜⎜ 1 Φ j + 2 Φ j −1 ⎟⎟ , β β ⎝β ⎠ ⎝β ⎠ Fψeff, j +1 = +

1

βτ

1 ~T 1 R uψ ⋅ (2U j − U j −1 ) − 2 Mψψ ⋅ (2Ψ j − Ψ j −1 ) + 2

βτ

βτ

RTuψ ⋅ [(2γ − 1)U j + (1 − γ )U j −1 ] −

1

βτ

Cψψ ⋅ [(2γ − 1)Ψ j + (1 − γ )Ψ j −1 ] +

β ⎛β ⎞ + Kψψ ⋅ ⎜⎜ 1 Ψ j + 2 Ψ j −1 ⎟⎟ . β ⎝β ⎠ eff

The effective matrix of stiffness K in Equation (A.33) has block saddle structure. This matrix could be factorized by using the square root method, that allows one to solve in each time layer only LSAE with lower and upper triangular matrices. Note, that the very important procedures, for example the torsions of the node degrees of freedom and the account of main boundary conditions which are necessary to form the final finite-element systems may be also realized in the symmetric forms without violation of structure of the FEM matrices. Finally, we point some features of modal analysis in ACELAN. This analysis is used to state the natural and resonance frequencies of elastic solids or the

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169

frequencies of electric resonances and antiresonances for piezoelectric solids. The frequencies of electric resonances f R = ω R / (2π ) and antiresonances

f A = ω A / (2π ) (where ωR and ωA are the corresponding circular frequencies) to be important characteristics of piezoelectric devices. They are also important in non-stationary analysis, for example in determination of damping factors. These frequencies are the natural ones of piezoelectric devices and calculated without account of influence of the damping and interaction with acoustic media. Thus, these frequencies could be estimated by FEM as a result of fulfillment of the modal analysis from solution of the generalized problems on eigenvalues based on Equations (A.26) – (A.29) without account of acoustic media

K ⋅ a = ω 2M ⋅ a ,

⎛M M = ⎜⎜ uu ⎝ 0

0⎞ ⎛K ⎟⎟ , K = ⎜ *uu ⎜K 0⎠ ⎝ uϕ

(A.35)

⎧U ⎫ K uϕ ⎞ ⎟, a = ⎨ ⎬ . − K ϕϕ ⎟⎠ ⎩Φ ⎭

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It should be noted that the matrices

(A.36)

K uϕ and Kϕϕ at the definition of

frequencies of resonances and antiresonances are differ. Problem (A.35) could be presented in equivalent form by reducing the massless degrees of freedom of the electric potential Φ as

~ K uu ⋅ U = λ M ⋅ U , λ = ω 2 ,

(A.37)

~ -1 -1 ⋅ K Tuϕ ⋅ U . K uu = K uu + K uϕ ⋅ K ϕϕ ⋅ K Tuϕ , Φ = K ϕϕ

(A.38)

In order to solve the generalized problems on eigenvalues (A.35), (A.36) or (A.37), (A.38), it is applied the code of the programs of packet ARPACK, realizing a modification of Arnoldi method. In this case, there are used in ACELAN the solution procedures of LSAE, factorization of matrices and multiplication of matrices on vectors without transformation of original matrices, but the matrix

-1 K ϕϕ is not calculated in evident form.

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A.2. TECHNIQUE OF SOLUTION OF THE DYNAMIC PROBLEMS IN ACELAN A.2.1. Construction of Finite-Element Model in ACELAN

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Consider construction of the geometrical model carried out in window ‘Plain Region’ (see Figures A.1 and A.2). In Figure A.1 are present actions on addition a point, namely (1) press buttons ‘Addition Object’, ‘Point’ and then click on image ‘Plain Region’, (2) the panel for reduction of the point coordinates. Analogous actions are fulfilled by adding edge (see Figure A.2), and it is necessary to keep the pressed left button of ‘mouse’ in the case of graphic addition.

Figure A.1. Addition of point.

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Figure A.2. Addition of edge.

A selection of the object is carried out by using cursor and pressing left button of ‘mouse’. In order to change the selected object, are used the buttons , and to remove the object − the button . A displacement of the selected object (see Figure A.3) is fulfilled by pressing buttons 1 with its trapping by ‘mouse’ and movement in the definite direction (see Figure A.3, left-hand) or by using the actions presented in Figure A.3, right-hand. In this case, all selected points will move. The action on addition of rectangular and ellipse are shown in Figures A.4 and A.5. In Figure A.4, the addition of the rectangular is carried out by pressing button 1, and two opposite tips of the rectangular are defined in panel 2. The rectangular may be also constructed graphically by movement of ‘mouse’ with keeping its pressed left button. In Figure A.5, the addition of the ellipse is carried out by pressing button 1, but the coordinates its centre − 2, the sizes of half-axes − 3 and number of the approximation points − 4 are introduced in panel ‘Ellipse’.

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All of the above actions and also the next ones may be carried out by using the commands of ACELAN-consol. The command list may be present into the consol window by using command ‘Help’. The definition of point by using the command is present in Figure A.6, where the call of the consol − 1, and definition of the point − 2 are shown.

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Figure A.3. Displacement of point.

Figure A.4. Addition of rectangular.

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Figure A.5. Addition of ellipse.

Figure A.6. ACELAN-consol.

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The next step of the problem solution is determination of boundary conditions at the edges and conditions of fastening and loading at the nodes. These operations are also fulfilled into window ‘Plain Region’ (see Figure A.7) which presents half part of axial section of the piezoelement and panel of determination of the boundary conditions and fastening at the nodes. Note, that in order to define antiresonance frequencies in one of the electrodes, it is necessary to state flag ‘Charge’ instead of ‘Electric Potential’. Then, the materials and degrees of freedom are found into window ‘Subregions’ (see Figure A.8), and a triangulation of the regions carried out into window ‘Region Division’ (see Figure A.9). An important feature of the software is possibility of packet regime for the problem solution. With this aim, the problem as whole or its part is described by commands in text file which is initiated by button ‘From File’ on the ACELANconsol (see Figure A.6).

Figure A.7. Definition of boundary conditions.

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Figure A.8. Selection of materials.

Figure A.9. Division of the region.

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A.2.2. Definition of Natural Frequencies of Resonance and Antiresonance Then, we go to program-mark ‘Solution’ (see Figure A.10), state solver ‘Natural Frequencies for Axes-Symmetric Region’ − 1 and select its tuning − 2. In the case of definition of the antiresonance natural frequencies, it is stated a sufficient distribution of memory − 3 in the solver parameters. Into window ‘Postprocessor Processing of Results’, the results of modal analysis are pointed (see Figure A.11). In order to exam the stress-strain characteristics, it is necessary to select natural frequency − 1, state the postprocessor parameters and initiate its − 2, select a distribution of the regarded stress-strain state parameter − 3, define character of image and scale reflection − 4, 5. Table A.1 shows first six natural frequencies of resonance

fR

and antiresonance

fA

which are present in screen

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and file.

Figure A.10. Solver tuning.

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177

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Figure A.11. Postprocessor window.

Table A.1. Natural frequencies of resonance and antiresonance Number 1 2 3 4 5 6

fR 24.383 88.320 143.74 152.53 224.17 254.27

fA 24.765 89.078 148.74 156.37 224.31 255.38

As a result of fulfillment of the modal analysis, it is estimated a spectrum natural frequencies ω с and corresponding eigenforms of the oscillations. The examples of application of the complex ACELAN are present in Figures A.12 and A.13, respectively for obtained forms of longitudinal contour oscillation mode of thin square plate and shear oscillations excited into plate with thickness polarization.

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Figure A.12. Forms of longitudinal contour oscillation mode of thin square plate.

Figure A.13. Forms of shear oscillations excited into plate with thickness polarization.

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179

A.2.3. Solution of Problem on Steady Oscillations The steady oscillations of piezoelement are excited by voltage on its electrodes, which is determined into window ‘Plain Region’. In this case (see Figure A.7), no zero boundary conditions are defined, for example voltage 200 V. The damping factors α , β , ζ d presented into Equations (A.1) − (A.3) may be estimated on the base of qualities Q1 , Q2 in two resonance frequencies

f R1 , f R 2 at the condition that Q1 = Q2 = Q by using the approximation formulas:

α=

2π f R1 f R 2 , Q ( f R1 + f R 2 )

β =ζd =

1 . 2π Q ( f R1 + f R 2 )

(A.39)

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In the case of the quality equal to Q = 350 , these factors for second and third resonance frequencies have values of α = 982.1 and β = ζd = 0.196×10−8. The computed values of the damping factors should be introduced into data base for corresponding materials (see Figure A.14). Then, after triangulation, we go to program-mark ‘Solution’ (see Figure A.15), select solver ‘Steady Oscillations − Axes-Symmetric Problem’ − 1, define frequency range − 2 and start solution. After its finish, we select the oscillation frequency − 3 and the stress-strain parameter − 4 into this window. In order to construct the amplitude-frequency characteristic (AFC), it is necessary to go to window of finite-element model ‘Region Division’ (see Figure A.16) and click by ‘mouse’ in this window. After that, panel ‘Time Processor’ − 1 appearances, and it is need to select stress-strain characteristic − 2 into this panel, introduce node number − 3 or click on its in FE model − 4. In Figures A.17−A.19, the constructed AFCs of vertical displacement (respectively, real and imagine parts, and also its module) of the central point (node No 5) at the face surface of the piezoelement are present.

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Figure A.14. Editor of materials.

Figure A.15. Solution and examination of results of the harmonic analysis.

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Figure A.16. Construction of amplitude-frequency characteristic.

Figure A.17. ACF of real part of vertical displacement for node No 5.

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Figure A.18. ACF of imagine part of vertical displacement for node No 5.

Figure A.19. ACF of module of vertical displacement for node No 5.

A.2.4. Solution of Non-stationary Problem Regard an impulse excitation of oscillations of the piezoelement due to the step-wise on time potential ϕ (t ) = V0 [ H (t ) − H (t − t 0 )] (where H (t ) is

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Heviside’s function, V0 = 200 V, t 0 = 1.0 × 10 −5 s) applied to lower electrode at the earthed upper electrode. The piezoelement is fastened on external radius of upper face. By taking into account the above pointed steps of FE model, we note only additional actions which are necessary to solve the non-stationary problem. First, before statement of the boundary conditions depending on time, it is necessary to select the problem type (‘Non-stationary Axes-Symmetric Problem’ − 1 in the considered example) into window ‘Postprocessor Processing of Results’ (see Figure A.20). Then, by pressing button ‘Selection of Ranges’ − 2, it is defined their number. By this, the range bounds and integrating step or number of subranges may be found or changed before immediate call of solver.

Figure A.20. Tuning of time ranges.

After that, it is necessary to return into window ‘Plain Region’ (see Figure A.21), select corresponding boundary − 1, define amplitude value of the corresponding component and click by ‘mouse’ on text with mark (this text distinguishes in panel a red strip − 2). Then, by pressing button ‘Function’, we go in panel ‘Function in Range’ and state in its demanded time range − 3 (one is first range in our example). Further, the functional dependence is found − 4 (here one is constant equal to one) and carryied out confirmation ‘OK’ by pressing button

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‘Apply’. This procedure should be repeated for all ranges and boundary conditions depending on time. The zero boundary conditions are selected automatically.

Figure A.21. Definition of boundary condition depending on time.

After that, by carrying out the above described steps (the material selection and triangulation), we can go to call of solver of the non-stationary problem. Additionally note some possibilities of time postprocessor. After solution of the problem, it is automatically activated window ‘Postprocessor Processing of

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Results’ (see Figure A.22). In Figure A.22, figure 1 signs the button of parameters of the postprocessor, figure 2 is the button of start of the postprocessor, figure 3 defines the time for imagine of fields, figure 4 selects the component σ rr of the mechanical stress tensor which is shown in the deformed state of the region at the

t = 9.0 × 10 −6 s. Analogously, the stresses σ θθ , σ zz , σ rz are constructed which present in Figures A.23 – A.25, respectively. Into this window, there is also possibility to remain data in file and print the imagine by using the buttons

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, respectively. Moreover, there is possibility to change the palette including ‘grey tones’ (see Figure A.26).

Figure A.22. Window of imagine of fields.

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Figure A.23. Distribution of stress

σ θθ .

Figure A.24. Distribution of stress σ zz .

Figure A.25. Distribution of stress σ rz .

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Figure A.26. Possibility to change the palette of imagine.

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In order to construct amplitude-time characteristics (ATC) of components of the stress-strain state, it is necessary to go into window ‘Region Division’ and by clicking ‘mouse’ to select in panel ‘Time Processor’ the interesting component and node number. By this, the node may be selected by using ‘mouse’ in FE and node model presented in this window. For example, Figure A.27 shows ATC of vertical displacement of the central point in upper face of piezoelement. The displacement is maximal at the time t = 9.0 × 10 −6 s for which the above distributions are present.

Figure A.27. Amplitude-time characteristic of displacement.

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[114] Rudyak, V. M. In Actual Problems of Modern Physics of the Ferroelectric Phenomena; Rudyak, V. M.; Ed.; Kalinin State University Press: Kalinin, 1978; pp 71 – 102. [115] Russian Standards (GOST) No 8.207-76. Direct Measurements with Multitime Observations. Processing Methods of Observation Results; Gosstandart: Moscow, 1976; pp 1-21. [116] Russian Standards (GOST) No 12370-85. Piezoceramic Materials. Test Methods; Gosstandart: Moscow, 1981; pp 1 – 10. [117] Rybjanets, A. N.; Nasedkin, A. V.; Turik, A. V. Integr. Ferroelectrics. 2004, Vol. 63, 179 – 182. [118] Senik, N. A. Strength Mater. 1983, No 4, 23 – 26. [119] Serebrennikov, M. G.; Pervozvansky, A. A. Discovery of Closed Periodicities; Nauka: Moscow, 1965; pp 1-244. [120] Shevtsov, S. N.; Akopyan, V. A.; Bragin, S. A. J. Mater. Meth. Techn. 2007, Vol. 1, 70 – 79. [121] Shevtsov, S. N.; Akopyan, V. A.; Soloviev, A. N., Chinchyan, L. V. In Theory and Practice of Manufacture Technology of Goods from Composite Materials and New Metallic Alloys; Moscow State University Press: Moscow, 2007; pp 436-442. [122] Shevtsov, S. N.; Soloviev, A. N.; Akopyan, V. A.; et al. In Proc. South. Sci. Centre Russian Acad. Sci.; Matishov, G. G.; Ed.; SSC RAS: Rostov-onDon, 2007; Vol. 2, pp 149-179. [123] Shubnikov, A. V.; Zheludev, I. S.; Konstantinova, V. P.; Sil’vestrova, I. M. Investigation of Piezoelectric Textures; USSR Academy of Sciences Press: Moscow, 1955; pp 1-184. [124] Shul’ga, N. A.; Bolkisev, A. M. Oscillations of Piezoelectric Solids; Naukova Dumka: Kiev, 1990; pp 1-228. [125] Shvedova, L. A. Device for Tests of Piezoelement by Pulsing Force. USSR Patent No 447873; Gospatent: Moscow, 1974; pp 1-27. [126] Smazhevskaya, Ye. T.; Fel’dman, N. B. Piezoelectric Ceramics; Soviet Radio: Moscow, 1971; pp 1-141. [127] Solokhin, N. V. Methods of Plane Dynamic Problems. PhD Dissertation; Rostov State University Press: Rostov-on-Don, 1984; pp 1-117. [128] Stepnov, M. M. Statistical Methods of Processing of the Mechanical Test Results; Mashinostroenie: Moscow, 1985; pp 1-231. [129] Tiersten, H. F. Linear Piezoelectric Plate Vibration; Plenum Press: New York, 1968; pp 1-211.

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[130] Timoshenko, S. P.; Parfenov, B. G. Device for Measurement of Resonance and Antiresonance Frequencies of Piezoceramic Resonators. USSR Patent No 742822; Gospatent: Moscow, 1980; pp 1-11. [131] Turik, A. V., Hasabova, G. I. In Semiconductors – Ferroelectrics; Kramarov, O. P.; Ed.; Rostov State University Press: Rostov-on-Don, 1978; pp 43-47. [132] Turik, A. V.; Komarov, V. D. Izv. USSR AS. Series Phys. 1970, Vol. 34(12), 2623 – 2627. [133] Ulitko, A. F. In Thermal Stresses in Elements of Constructions; Shevchenko, Yu. N.; Ed.; Naukova Dumka: Kiev, 1975; Vol. 15, pp 90-99. [134] Ulitko, A. F. In Modern Problems of Mechanics and Aviation; Kudryavtsev, B. A.; Ed.; Moscow Aviation Institute Press: Moscow, 1982; pp 290 – 300. [135] Ultrasound Transducers; Kikuchi, Y.; Ed.; Corona Publ. Co.: San Antonio, TX, 1972; pp 1-406. [136] Vasil’chenko, C. Ye.; Nasedkin, A. V.; Soloviev, A. N. Comput. Techn. 2005, Vol. 10(1), 10 – 20. [137] Vatulyan, A. O.; Soloviev, A. N. J. Appl. Mech. Techn. Phys. 1999, Vol. 40(3), 204 – 210. [138] Wang, H.; Cao, W. J. Appl. Phys. 2002, Vol. 92(8), 4578 – 4583. [139] Yakovlev, L. A.; Serebrennikov N. P. Russian J. Nondestruct. Test. 1980, No 7, 52 – 57. [140] Yakovlev, L. A.; Zobnin, L. P. Ultrasound Method of Definition of the Piezoelectric Constants in Piezoceramics. USSR Patent No 530279; Gospatent: Moscow, 1976; pp 1-9. [141] Yanovsky, V. K.; Keshishyan, T. N. In Physical and Chemical Foundations of Ceramics; State Press of Literature on Buiding Materials: Moscow, 1956; pp 546-549. [142] Zinchenko, V. N.; Kandyba, P. Ye.; Mezheritsky, A. V. Electronic Techn. 1981, Vol. 4(94), 30 – 35. [143] Zobnin, L. P.; Nesmashnaya, O. M.; Yakovlev, L. A. Izv. LETI. 1974, Vol. 145, 78 – 83.

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INDEX

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A  accuracy, 43, 50, 51, 54, 58, 78, 99, 100, 101, 108, 117, 120, 162 ACF, 181, 182 achievement, 139, 145 acoustic waves, 17 acoustical, 1 additives, 139, 149 adiabatic, 12, 14 AFC, 5, 60, 61, 62, 66, 68, 69, 71, 72, 73, 74, 75, 78, 93, 179 AFCs, 6, 79, 80, 82, 83, 87, 98, 179 aggregates, 3 aging, 112 air, 20, 21, 31 Aircraft, 189 algorithm, xii, 6, 53, 93, 94, 101, 104, 119, 129 alternative, 44, 167 aluminum, 51 amplitude, 5, 26, 27, 28, 49, 54, 60, 67, 73, 78, 80, 93, 129, 140, 152, 153, 179, 181, 183, 187 analog, 161 annealing, 94 antenna, 1 appendix, 4 application, xi, 50, 53, 54, 108, 117, 131, 139, 156, 177

argument, 78 arithmetic, 67, 166 aspiration, 65 assumptions, 33 asymptotic, 65 ATC, 187 averaging, 40, 167

B  barium, 6, 74 beams, 50 behavior, 1, 57, 113, 114 bending, 4, 6, 50, 54, 55, 61, 79, 119, 120, 121, 126, 129, 131, 135, 137, 138, 140, 141, 142, 143, 145, 150, 151, 152, 153, 154, 155, 156 bismuth, 7 blocks, 166 boundary conditions, 19, 20, 21, 22, 24, 25, 26, 30, 31, 32, 34, 37, 38, 88, 161, 162, 163, 168, 174, 179, 183, 184 boundary surface, 163 bounds, 20, 84, 183 buffer, 108 burning, 143 buttons, 170, 171, 185

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200

Index

C 

D 

calibration, 117 capacitance, 39, 52, 53, 92, 101, 107, 108, 116 cell, 93, 109 ceramic, 52, 74, 94, 96, 99, 100, 112, 114, 133, 142, 143, 149, 154, 159 ceramics, 4, 5, 55, 64, 78, 79, 119, 127, 135, 137, 138, 139, 144, 145, 147, 148, 149, 154, 159 channels, 109 charged particle, 9 chemical composition, 139 classification, 49, 125 Co, 193, 197 compatibility, 2, 5, 54, 59, 60, 101 compensation, 76 compliance, 13, 14, 17, 29, 36, 43, 44, 45, 46, 84, 88, 97, 99, 102, 103, 133, 139, 151, 156, 157 components, xii, 4, 9, 10, 11, 12, 13, 15, 16, 17, 19, 37, 44, 56, 99, 116, 124, 134, 162, 163, 187 composites, xii composition, 6, 7, 52, 53, 56, 72, 74, 78, 87, 100, 101, 114, 139, 144, 149 computation, xii, 2, 7, 80, 81, 93, 99, 119 concrete, 19 conductivity, 28, 29, 30, 60, 61, 78, 144 configuration, 4 conjugation, 165 construction, 4, 5, 49, 80, 104, 111, 145, 170 construction materials, 49 continuity, 20, 161, 163 control, 1, 4, 130, 131 convergence, 98 cooling, 134, 136, 137 correlation, 99, 146 couples, 146 coupling, 3, 46 covering, 7, 21, 40 critical value, 157 crystalline, 9, 10, 14 crystals, 10, 11, 139

damping, 4, 49, 53, 93, 160, 161, 164, 169, 179 decay, 161 decomposition, 161 defects, 3, 131 definition, xi, xii, 2, 3, 4, 6, 28, 34, 35, 49, 50, 51, 52, 53, 54, 55, 60, 61, 65, 89, 94, 98, 105, 111, 116, 119, 121, 128, 129, 139, 154, 162, 169, 172, 176 deformation, 11, 49, 50 degrees of freedom, 164, 166, 168, 169, 174 density, 9, 26, 43, 86, 92, 93, 95, 96, 99, 143, 147, 160, 161 Department of Education, xii dependent variable, 12 depolarization, 55, 56, 134, 136, 144, 146 deposition, 89, 92 deviation, 102, 129, 154, 155, 156 dielectric constant, 12, 52, 56, 80 dielectric permeability, 2, 4, 12, 13, 15, 16, 17, 31, 32, 38, 53, 92, 101, 102, 103, 104, 161 dielectrics, 9, 21, 56 diffraction, 57 diffusion, 53 dipole, 9, 10 dipole moment, 9, 10 Discovery, 196 discretization, 163 discs, 45, 78 dispersion, 53, 154, 155 displacement, 5, 10, 19, 24, 26, 28, 33, 35, 38, 40, 44, 57, 67, 68, 69, 73, 89, 92, 130, 163, 164, 171, 179, 181, 182, 187 distribution, 74, 154, 155, 157, 158, 176 division, 35, 57, 69, 163 domain structure, 55, 56, 113, 134, 137, 138, 139, 143, 144, 147, 149 domain walls, 139, 143 durability, 145 duration, 55, 131

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Index

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E  Education, xii elastic constants, 4, 6, 15, 44, 49, 50, 51, 54, 55, 59, 88, 95, 99, 100, 101, 102, 103, 119, 131, 137, 138, 144, 146, 147, 151, 152, 154, 157 elasticity, 2, 4, 6, 17, 19, 20, 53, 80, 119, 129, 131, 132, 133, 147, 159 electric charge, 11, 20, 56, 74, 76, 106, 107, 108, 109, 110, 115, 144, 162 electric circuit, 162 electric conductivity, 61, 78 electric current, 19, 20, 108 electric energy, 19, 47 electric field, 4, 9, 10, 11, 12, 13, 16, 17, 20, 21, 29, 31, 32, 37, 41, 43, 47, 55, 56, 57, 58, 60, 69, 73, 75, 77, 88, 89, 90, 92, 106, 139, 144, 153, 154, 160 electric potential, 16, 19, 20, 21, 24, 26, 29, 37, 41, 161, 162, 164, 169 electrodes, 3, 19, 20, 21, 22, 23, 25, 26, 30, 31, 34, 37, 38, 40, 52, 58, 76, 89, 92, 95, 106, 112, 131, 143, 144, 162, 174, 179 electromagnets, 129 elongation, 50 energy, 4, 11, 19, 46, 47, 119, 120, 122, 123, 134 engines, 3 entropy, 12 equality, 37, 67, 143, 162 equating, 39 equilibrium, 161 excitation, 40, 49, 51, 129, 153, 182 exclusion, 107 expert, vi exploitation, 107 exposure, 131, 136, 137 external influences, 134, 144

F  failure, 134 family, 6

201

FEM, 53, 80, 168, 169 ferroelectrics, 10, 51, 54, 55, 57, 105, 139, 147 filters, xi financial support, xii flow, 56 fluctuations, 101, 103, 104 freedom, 155, 157, 164, 166, 168, 169, 174 fulfillment, 43, 93, 169, 177

G  gases, 159 genetic algorithms, 159 Gibbs, 12, 106 graduate students, xii grain, 94, 99 grants, xii groups, 11, 23, 49, 50, 56, 57, 62, 69 growth, 99, 112, 113, 115, 144, 152, 154

H  harmonics, 43, 62, 69, 76 health, 3 heat, 12, 46, 49, 50, 53, 56, 134, 136, 137, 144, 145 heat loss, 53 height, 60, 61, 69, 72, 74, 77, 87, 98, 114, 115, 116, 117, 128 heterogeneity, 2, 52, 107, 111 heterogeneous, 53, 144, 152, 154, 159 high temperature, 54, 129 high-frequency, 53 Holland, 2, 193 homogeneity, 111, 114 hydro, 1 hypothesis, 67, 157 hysteresis, 49

I  identification, xii, 4, 5, 6, 44, 59, 80, 87, 146 identification problem, 6

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Index

identity, 82 immersion, 53 in transition, 59, 76 inactive, 37 incompatibility, 52, 95 independent variable, 4, 12, 13, 14, 16, 21, 32, 37, 41 indices, 4, 13, 18 induction, 12, 17, 20, 32, 37, 88, 106, 139, 160, 162 industry, 1 inertia, 76, 122, 126, 128, 156 infinite, 4, 11, 14, 161 initiation, 10, 11, 74, 94 injection, 144 injury, vi integration, 34 interaction, 9, 47, 169 interface, 20 Investigations, 189, 194 ions, 10, 143 isotropic, 2, 4, 11, 14, 15, 37, 41, 49, 54, 88, 125, 146 isotropic media, 15 isotropy, 18, 126 iteration, 166

K  kinetic energy, 120, 122, 123

L  language, 159 lanthanum, 149 lattice, 9, 10, 147, 149 law, 157, 160 limitations, xi linear, 2, 4, 11, 17, 30, 46, 57, 65, 106, 117, 139, 140, 159, 160, 161, 166 linear dependence, 117 liquidate, 115 liquids, 159 London, 191, 192, 193, 195

losses, 29, 35, 39, 53, 54, 57, 107, 131, 161 low temperatures, 50

M  magnetic effect, 17 magnetic field, 17 matrix, 4, 5, 6, 13, 14, 15, 16, 32, 54, 88, 93, 95, 99, 104, 122, 123, 124, 164, 165, 166, 168, 169 Maxwell equations, 20 mean-square deviation, 154, 155, 156 mechanical properties, 4, 145 mechanical stress, 4, 10, 11, 13, 16, 17, 19, 21, 32, 41, 57, 74, 76, 106, 111, 112, 113, 115, 134, 137, 144, 152, 154, 160, 162, 185 media, 4, 7, 12, 15, 20, 53, 159, 161, 169 MEK, 193 memory, vii, 176 metals, 143 microscope, 129, 131 microscopy, 95 microstructure, 6, 95, 99 mobility, 139, 144 modeling, 4, 161 models, xi, 2, 3, 4, 7, 139, 159 momentum, 9, 10, 121, 122, 126, 128, 156 Moscow, 189, 190, 192, 193, 194, 195, 196, 197 motion, 17, 21, 23, 32, 33, 75, 116, 120, 124, 159, 160, 161 mouse, 170, 171, 179, 183, 187 movement, 17, 171 MRA, 2, 3, 5, 50, 51, 52, 53, 54, 55, 119, 150, 152, 154, 155, 156 MRS, 51, 55 multiples, 75 multiplication, 169

N  natural, 69, 112, 120, 126, 131, 144, 146, 168, 176, 177 neglect, 17, 116, 126

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Index New York, 191, 193, 194, 195, 196 nodes, 164, 174 non-destructive, 1 non-linearity, 57, 117, 140 normal, 19, 20, 21, 22, 40, 154, 157, 158, 162, 163

O  ordinary differential equations, 7, 164 orientation, 56, 57, 58 oxide, 7

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P  packets, 161 parameter, 22, 49, 107, 176, 179 particles, 9 Pb, 97 PCM, 1, 2 PCR, 6, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103 pendulum, 49, 54 permeability, 2, 4, 12, 13, 15, 16, 17, 31, 32, 38, 53, 92, 95, 101, 102, 103, 104, 161 permit, xi perovskite, 146, 147 perturbations, 101 phase transitions, 147 physical properties, 2, 95 Piezoceramics, ix, 94, 119, 197 piezoelectric properties, 11, 15, 111, 114, 159 piezoelectricity, 3 planar, 5, 39, 40, 41, 46, 47 pleasure, xii Poisson, 41, 42, 46, 50, 51, 91, 147, 148, 149 Poisson ratio, 42 polarizability, 56, 146 polycrystalline, 10, 105 polymer, xii polymers, 54 polynomial, 145, 146 polynomials, 44, 145 porosity, 99

203

porous, 159 potential energy, 122 power, 35, 117 pressure, 32, 161 probability, 68, 117, 154, 156, 158 program, 5, 80, 145, 176, 179 propagation, 17, 26, 161 property, vi, 50, 87

Q  quantitative estimation, 66, 146, 149 quartz, 53, 76, 79, 144

R  radio, 1 radius, 183 range, xi, 3, 6, 10, 49, 51, 53, 55, 66, 68, 69, 78, 82, 115, 116, 117, 119, 127, 130, 133, 134, 138, 139, 142, 143, 145, 146, 147, 149, 154, 155, 179, 183 RAS, 196 Rayleigh, 161 redistribution, 74 reflection, 176 regular, 144 relationship, 6, 10, 11, 17, 29, 30, 34, 35, 38, 39, 43, 44, 45, 46, 47, 55, 67, 76, 101, 106, 109, 125, 126 relationships, 5, 6, 19, 23, 24, 33, 42, 43, 53, 57, 66, 67, 98, 99, 120, 121, 125, 127, 128, 139, 146, 160, 161, 164 relative size, 78 relaxation, 55, 114 reliability, 2, 6, 60, 74, 80, 98, 99, 100, 117 research and development, xii, 3, 144 resistance, 108 room temperature, 119, 130, 134, 136, 152 roughness, 112, 114 Russia, 4, 159

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204

Index

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S  sampling, 95, 96, 97, 100, 102, 112, 131, 132, 133, 138, 141, 147, 148, 152, 153, 154, 155, 157 satellite, 4 satisfaction, 59 scaling, 49 scattering, 31, 53, 54, 59, 154 search, 25 sensitivity, 3, 58 sensors, xi services, vi shape, 2 shear, 5, 6, 36, 37, 38, 39, 40, 43, 50, 52, 59, 65, 73, 87, 89, 92, 93, 95, 98, 99, 119, 128, 129, 131, 132, 133, 135, 136, 138, 157, 177, 178 shortage, 54 sign, 27, 57, 100, 101, 115, 125, 135, 140, 142, 153 signals, 1, 110 significance level, 158 signs, 10, 33, 43, 74, 76, 83, 146, 185 silver, 143 simulation, 5, 80 sintering, 2, 3, 59, 94 sites, 65, 114, 162 smoothing, 99 sodium, 7 software, 80, 174 spatial, 9 spectrum, 5, 74, 93, 177 stability, 71, 113, 131, 134, 139, 145 stabilize, 145 Standards, 51, 93, 128, 193, 196 steady state, 107 steel, 117 stiffness, xii, 42, 43, 93, 99, 121, 122, 124, 131, 143, 144, 145, 164, 168 strain, 7, 11, 12, 16, 17, 37, 50, 92, 119, 125, 134, 137, 138, 154, 176, 179, 187 strains, 16, 21, 50, 160 stress, 7, 11, 13, 17, 32, 37, 40, 76, 92, 106, 107, 111, 113, 114, 115, 119, 129, 137,

138, 152, 154, 161, 162, 163, 176, 179, 185, 186, 187 strontium, 6 structural health monitoring, 3 students, xii substitution, 4, 13, 17, 75, 91, 124, 126, 128, 131, 154, 166 superimpose, 98 superposition, 80, 83, 86, 116 supply, 19 surface layer, 143, 144 susceptibility, 56 switching, 55 symmetry, 2, 4, 11, 14, 15, 18, 39, 88, 106, 121, 129, 138, 162, 163, 165

T  temperature, 5, 10, 12, 50, 51, 55, 56, 116, 117, 119, 127, 130, 131, 132, 133, 134, 136, 137, 138, 139, 141, 142, 143, 144, 145, 146, 147, 148, 149, 157 temperature dependence, 134, 145 tension, 50, 89, 91, 93, 126, 152, 153 test data, 62, 64, 146, 147, 152, 154 thermal expansion, 126, 127 thermal load, 131 thermal stability, 134 thermal treatment, 12, 51, 55, 139 thermodynamic, 4, 11, 12 transducer, 57, 110, 146 transfer, 123 transformation, 40, 169 transformations, 93 transition, 24, 50, 51, 52, 98, 107, 139, 145, 146, 149, 163 transition temperature, 50 transitions, 147 transmission, 109, 110 triangulation, 163, 174, 179, 184

U  ultrasound, 3, 53

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Index uncertainty, 98 uniform, 152 USSR, 111, 117, 189, 190, 193, 195, 196, 197

V 

78, 89, 90, 121, 123, 127, 138, 152, 160, 161, 162, 163, 164, 166 velocity, 26, 41, 62, 79, 95, 122, 123, 161 vibration, 5 viscosity, 53

W  wave number, 26, 27, 28, 75 wave propagation, 37 witnesses, 98, 149 workers, xii WSF, 1

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vacuum, 50, 92 validity, 5, 7, 62, 88, 99, 100, 101, 102, 104, 111, 146, 147, 149 variables, 12, 24, 33 variation, 108, 154 vector, 9, 10, 11, 17, 19, 20, 21, 22, 25, 28, 37, 38, 39, 52, 56, 61, 66, 67, 68, 72, 77,

205

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