Piezoceramic Materials and Devices [1 ed.] 9781617289583, 9781608764594

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Piezoceramic Materials and Devices, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Piezoceramic Materials and Devices, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

MATERIALS SCIENCE AND TECHNOLOGIES

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PIEZOCERAMIC MATERIALS AND DEVICES

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MATERIALS SCIENCE AND TECHNOLOGIES

PIEZOCERAMIC MATERIALS AND DEVICES

IVAN A. PARINOV

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EDITOR

Nova Science Publishers, Inc. New York

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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 Piezoceramic materials and devices / Ivan A. Parinov, editor. p. cm. Includes index. ISBN 978-1-61728-958-3 (e-book) 1. Piezoelectric ceramics. 2. Piezoelectric devices. I. Parinov, Ivan A. TK7871.15.C4P54 2009 620.1'4--dc22 2009044329



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DEDICATION To memory of academician

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Iosiph Izrailevich Vorovich

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CONTENTS Preface Chapter 1

Lead-Free Long Ago L. A. Reznichenko, I. A. Verbenko, K. P. Andryushin, S. I. Dudkina and V. P. Sakhnenko

Chapter 2

High Performance of Advanced Composites Based on Relaxor-Ferroelectric Single Crystals V.Yu. Topolov, C.R. Bowen and S.V. Glushanin

71

Ceramic Piezocomposites: Modeling, Technology, and Characterization A.N. Rybyanets

113

Some Finite Element Methods and Algorithms for Solving Acoustopiezoelectric Problems A. V. Nasedkin

177

Chapter 3

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ix

Chapter 5

Chapter 6

Chapter 7

Identification of Effective Properties of the Piezocomposites on the Basis of Finite Element Method (FEM) Modeling with ACELAN A. N. Soloviev and G. D. Vernigora Toughening Mechanisms and Fracture Resistance of Ferroelectric Materials I. A. Parinov Active and Passive Vibration Control of Aircraft Composite Structures Using Power Piezoelectric Patch-Like Actuators S. N. Shevtsov and V. A. Akopyan

Index

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285 325

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PREFACE Piezoelectric materials and composites and also the devices based on them have found wide applications in the modern technologies. This new book drives the reader from the concepts and ideas in this scientific area to theoretical, model and experimental results obtained some internationally recognized research teams. It is a collection of seven comprehensive overview papers prepared by leading scientists in Russia in order to collect the involved expertise in the area of novel piezoelectric materials, composites and devices and also methods of their investigation. Chapter 1 - The basis of the piezoceramic ferroelectric materials most commonly used by modern industry is solid solutions of lead-containing systems, such as PbTiO3 – PbZrO3, Pb(Nb2/3Mg1/3)O3 – PbTiO3, and the like. The technologies most commonly used in the creation of piezoceramic ferroelectric materials involve solid-phase synthesis and sintering at high temperatures. Due to the significant toxicity of lead compounds, alternative materials and methods without the use of lead - have been intensively sought after in the recent years. An inventory is presented of ecologically safe materials and methods for use in electrotechnical industries and products. The results as of 1970 indicate suitable lead-free materials containing compounds of many other structural types, such as perovskites (sodium niobate and potassium niobate); pseudoelements (lithium niobate and lithium tantalite); multilayer perovskite-like compounds (strontium pironiobate and calcium pironiobate), columbites (CdNb2O6, CaNb2O6, SrNb2O6), and others. Phase diagrams of binary and multicomponent systems of these objects describe their dielectric, piezoelectric, and elastic properties under a broad range of external influences (temperatures, frequencies of alternating field and strength of constant field). The theoretical foundations and empirical rules of these compounds for use in basic components for lead-free piezoelectric materials are also described. Chapter 2 - The outstanding electromechanical properties of single crystals of relaxorferroelectric solid solutions near the morphotropic phase boundary suggest that these materials can be regarded as potential components of high-performance piezo-active composites. In this chapter the authors discuss the advantages of using the relaxorferroelectric single crystals of solid solutions of (1 – x)Pb(Mg1/3Nb2/3)O3 – xPbTiO3 and (1 – y)Pb..(Zn1/3Nb2/3)O3 – yPbTiO3 as components of single crystal / polymer and single crystal / porous polymer composites of the 2–2 and 1–3 types. Examples of the high piezoelectric

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Ivan A. Parinov

activity and sensitivity and large hydrostatic parameters of the aforementioned composites are analysed in connection with the electromechanical properties of the single-crystal component. The influence of the polarisation orientation on the effective electromechanical properties and related parameters of the single crystal / polymer composite are discussed for different connectivity patterns. Comparisons are made betthe authorsen the effective parameters of 2–2 connectivity composites based on polydomain single crystals and single-domain single crystals of the same chemical composition (for instance, 0.67Pb(Mg1/3Nb2/3)O3 – 0.33PbTiO3). Specific effective parameters of single crystal / polymer composites are compared to those of piezo-active composites based on the conventional ferroelectric ceramic and having the same connectivity pattern (either 2–2 or 1–3). In general, data presented in the chapter show how the relaxor-ferroelectric single-crystal component improves the effective parameters of the novel composites and promotes the formation of non-monotonic volumefraction dependences of the effective parameters which are of particular interest for a variety of piezotechnical applications, such as active elements of sensors, actuators, transducers, hydrophones, etc. Chapter 3 - A comprehensive review of microstructure peculiarities, mathematical models, methods of fabrication and measurements, as well as systematic experimental data for different types of ceramic piezocomposites is presented. New families of polymer-free ceramic piezocomposites (porous ceramics, composites ceramics/ceramics and ceramics/crystals) with properties combining better parameters of PZT, PN type ceramics and 1-3 composites are introduced. Systematic experimental results for different porous piezoelectric ceramics with various connectivity types are presented. New ― damped by scattering‖ ceramic piezocomposites, characterized by previously unachievable parameter combinations are proposed. New material designing concept and fabrication methods for ceramic piezocomposites are considered. Piezoelectric resonance analysis methods for automatic iterative evaluation of complex material parameters, together with the full sets of complex constants for different ceramic piezocomposites, are presented. Critical comparison of FEM calculations of effective constants for the ceramic piezocomposites with the results of various approximated formulas, unit cell models and experimental data are carried out. Microstructural and physical mechanisms of losses and dispersion in ceramic piezocomposites, as well as technological aspects of its large-scale manufacture and application in ultrasonic devices are considered. Chapter 4 - This chapter considers mathematical modeling by FEM of the piezoelectric devices. The formulation of dynamic problems in compound domains with different physical properties (piezoelectric, elastic and acoustic) is given. A new generalized Kelvin model for damping inputs in piezoelectric analysis is proposed. This model is completely compatible with the mode superposition method. When semi-discrete FEM approximations of the solution are applied to the governing equations in weak form, variational FEM equations with symmetrical saddle matrices are derived. A set of algorithms using symmetrical saddle matrices to create and solve FEM equations is proposed for static and dynamic problems. The Newmark method without velocities and accelerations node values is used for step-by-step time integration scheme; and modified Chollessky decomposition method is used as linear system solver. All procedures needed in FEM manipulations (the degree of freedom rotations, mechanical and electric boundary condition settings, etc.) also are provided in symmetrical form. FEM for evaluation of natural frequencies of compound electroelastic bodies are investigated. The schemes

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Preface

xi

presented use FEM block matrices, where different matrix blocks are related to different field variables. The computer program ACELAN was developed based on these algorithms. The program was tested carefully and the results were compared with analogous values obtained by ANSYS, another well-known computer program. The numerical experiments showed that ACELAN and its algorithms are effective and give accurate results. Finally, new mathematical models and numerical methods for modelling the dynamic behaviour of three-dimensional piezoelectric devices with rotation effects are presented. For piezoelectric vibratory gyroscopes working on ― energy trapped‖ effects, the authors introduce a small parameter, which is the ratio between rotation frequency and the principal resonance frequency. For modelling the work of piezoelectric gyroscopes, the authors use a series expansion of this small parameter. In the first phase, for zero approximation, the authors solve the eigenvalue and harmonic problems close to resonance frequency. The mechanical displacement results are stored in the nodes of finite element mesh for use in the next step. In the second phase, the authors solve the problem with axial rotation and relative displacement for resonance frequency, where Coriolis forces are considered as body forces. The authors also obtain the formulae for Coriolis forces concentrated in the nodes of finite elements. Chapter 5 - The authors developed a method of defining effective elastic and piezoelectric properties of irregular structure composite piezoceramics. The method is based on dynamic equivalence. The authors examine the oscillations of some meaningful volumes of composite materials and find their resonance and antiresonance frequencies. The calculations are carried out in the finite element ACELAN complex using the calculation module for irregular structure composite materials. This work also considers the estimation algorithm of a full set of composite piezoceramics effective constants on basis of static tasks. The authors use the equations and results received after the ACELAN calculation of vertical and horizontal displacements and of the potential of meaningful volume of porous ceramics of irregular structure. The results of quantitative experiments are given below. Chapter 6 - Various toughening mechanisms associated with microstructure features of ferroelectric materials can lead to crack shielding, thus significantly increasing the effective fracture toughness and other strength properties in these materials. In other cases, the crack interaction with microstructure leads to the crack amplification. For this reason, it is important to study characteristic fracture mechanisms for ferroelectrics, define their main features, and couple the technological and microstructure peculiarities with properties of the prepared samples. This study first presents an overview of some methods and models of toughening mechanisms and fracture resistance related to ferroelectric materials. The discussion relates to considerations of microcracking and twinning zones near macrocracks, phase transformations, crack deflection, branching and bridging, and other phenomena. Then, a catastrophic crack growth condition is examined and the fracture energy changes are estimated in dependence on the presspowder initial porosity, and intergranular cracking and phase transformations in vicinity of the crack tip. A computer simulation is developed taking into account the actual physical model of gradient sintering for lead-zirconate-titanate (PZT) ceramics. The authors also present a fracture model and the results of its numerical realization for PZT ceramics, taking into account hysteretic ferroelectric domain-switching processes near the macrocrack tip. Both models are based on the balance of energy caused by driving forces and on a consideration of the total energy connected with the processes of dissipative interactions around the crack.

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Chapter 7 - This chapter presents some theoretical and real-world approaches to design and implementation of aircraft structures smart vibration control that is controlled by feedback and shunted by external circuits power PZT patches. First the authors consider the problem of vibration reduction in helicopter rotor blades, more particularly, the features of rotor blade dynamics and an approach to ensuring dynamic similarity between full scale and scaled rotor blade. On the basis of this analysis, the authors deduce the principal requirements for smart vibration control of rotor blades. One of the greatest technical difficulties of rotor blade active vibration damping is the necessity to transmit to a blade a number of high-voltage command signals through the rotated hub. The purpose of their investigation was to decrease the number of control channels while retaining good vibration damping efficiency. This study investigates the optimal type (i.e., bimorph or unimorph), location and sizes of plate-like actuators and sensors attached to the composite spar that bears the bend and twist load. The authors compare the working modes of active and passive (i.e., shunted by electric circuit) PZT actuators.. Numerical simulation shows that the passive damping mode is efficient in the high frequency range only. On other hand, with active control, the stability of control loop may be lost at some vibrations and feedback parameters. The authors propose an approach according to which all actively controlled PZT patches are driven at a narrow frequency band, filtering preferentially on a first eigenfrequencies. All installed shunted passive PZT patches will damp high vibration frequencies, while simultaneously increasing the stability of the control loop. Finally, the authors present some experimental results obtained for a scaled (1/7) rotor.

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Chapter 1

LEAD-FREE LONG AGO L. A. Reznichenko1, I. A. Verbenko, K. P. Andryushin, S. I. Dudkina and V. P. Sakhnenko Research Institute of Physics, Rostov State University, 194, Stachki Ave, Rostov-on-Don, 344090 Russia

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ABSTRACT The basis of the piezoceramic ferroelectric materials most commonly used by modern industry is solid solutions of lead-containing systems, such as PbTiO3 – PbZrO3, Pb(Nb2/3Mg1/3)O3 – PbTiO3, and the like. The technologies most commonly used in the creation of piezoceramic ferroelectric materials involve solid-phase synthesis and sintering at high temperatures. Due to the significant toxicity of lead compounds, alternative materials and methods - without the use of lead - have been intensively sought after in the recent years. An inventory is presented of ecologically safe materials and methods for use in electrotechnical industries and products. The results as of 1970 indicate suitable lead-free materials containing compounds of many other structural types, such as perovskites (sodium niobate and potassium niobate); pseudoelements (lithium niobate and lithium tantalite); multilayer perovskite-like compounds (strontium pironiobate and calcium pironiobate), columbites (CdNb2O6, CaNb2O6, SrNb2O6), and others. Phase diagrams of binary and multicomponent systems of these objects describe their dielectric, piezoelectric, and elastic properties under a broad range of external influences (temperatures, frequencies of alternating field and strength of constant field). The theoretical foundations and empirical rules of these compounds for use in basic components for lead-free piezoelectric materials are also described.

1 e-mail: [email protected].

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1. INTRODUCTION

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Most ferroelectric ceramic materials used in production at present are based on solid solutions (SS) of Pb-containing systems, such as PbTiO3 – PbZrO3 (PZT), Pb(Nb2/3Mg1/3)O3 – PbTiO3 (PMN-PT), and others. The conventional technology of their manufacturing includes a solid-state synthesis and sintering at high temperatures. For over 40 years, the Research Institute of Physics of Southern Federal University has intensively studied on the research, design, and experimental performance of such materials . With the assistance of the present authors, more than 30 thousand multicomponent complex oxides of different structural families were synthesize. and their crystallochemical properties were established. A new approach was worked out which enabled consideration of the problems of stability of structure and ferro-antiferroelectric states in these oxides from single positions. In addition, the regular relationships between composition, structure, and properties were determined, allowing the prediction of the characteristics of these novel materials. On this basis, a research methodology was developed. About 200 types of ferro-piezoelectric ceramic materials (FPCM) were engineered and the methods of their preparation were patented. These materials belong to 9 groups distinguished by a set of electrophysical parameters; and consequently the fields of their application ― overlap‖ in virtually all known piezotechnical directions. The nine groups of new materials are as follows: -

Group 1: Electrically and mechanically stable materials designated for devices operating in force regimes (piezotransformers, piezoelectric motors, ultrasonic radiators, high-voltage generators). These materials possess the following electrophysical parameters: TС = (505 ÷ 625) K, ε33T/ε0 = 900 ÷ 2300, Kp = 0.57÷ 0.66, ‫׀‬d31‫( = ׀‬85 ÷ 195) pC/N, ‫׀‬g31 ‫( =׀‬8.4 ’ 11.7) mV∙m/N, tgδ∙10-2 (E = 50 V/cm) = 0.3 ÷ 0.85, tgδ∙10-2 (E = 1 kV/cm) = 0.4 ÷ 1.0, Qm = 630 ÷2000, where TC is the Curie temperature, ε33T/ε0 is the relative dielectric permittivity of poled samples, Kp is the electromechanical coupling factor of the radial mode of vibration, ‫׀‬d31 eht si ‫׀‬ piezoelectric modulus, ‫׀‬g31 cirtceleozeip) tnatsnoc egatlov cirtceleozeip eht si ‫׀‬ response), tgδ is the dielectric loss tangent, and Qm is the mechanical quality factor.

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Group 2: Materials having a high dielectric permittivity designated for use in lowfrequency receiving devices (hydrophones, microphones, seismoreceivers). Their main electrical characteristics are as follows: ТC = (430 ÷ 555) K, ε33T/ε0 = 2800 ÷ 6000, Kp = 0.68 ÷ 0.71, K33 = 0.72 ’ 0.78, ‫׀‬d31‫( = ׀‬245 ÷ 380) pC/N, tgδ∙10-2 (E = 50 V/cm) = 1.2 ÷ 2.9, Qm = 35 ÷ 80, where K33 is the electromechanical coupling factor of the longitudinal mode of vibration. Group 3: High-sensitivity materials being effectively used in accelerometers, ultrasonic flaw detectors, devices for nondestructive control of objects by acoustic emission, and ultrasonic medical diagnostic equipment. Their main electrophysical characteristics are as follows: ТC = (620 ÷ 630) K, ε33T/ε0 = 650 ÷ 1400, Kp = 0.62 ÷ 0.68, ‫׀‬d31‫( = ׀‬95 ÷ 170) pC/N, ‫׀‬g31 ‫( =׀‬13.7 ÷ 16.5) mV∙m/N, tgδ∙10-2 (E = 50 V/cm) = 1.6 ÷ 2.0, Qm = 90 ÷ 105. Group 4: Materials with medium dielectric permittivity designated for transducers operating in receiving mode. They possess the following electrophysical

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characteristics: ТC = (590 ÷ 595) K, ε33T/ε0 = 1900 ÷ 2200, Kp = 0.65 ÷ 0.67, ‫׀‬d31‫= ׀‬ (205 ÷ 210) pC/N, ‫׀‬g31 ‫( =׀‬10.7 ÷ 12.1) mV∙m/N, tgδ∙10-2 (E = 50 V/cm) = 1.4 ÷ 1.6, Qm = 70 ÷ 75. Group 5: Materials with high anisotropy of piezoelectric parameters designated for ultrasonic flaw detectors, thickness gauges, medical diagnostic equipment, accelerometers, and piezoelectric indicators with abnormal sensitivity to hydrostatic pressure. Their main electrophysical characteristics are as follows: ТC = (600 ÷ 715) K, ε33T/ε0 = 120 ÷ 180, Kp = 0 ÷ 0.07, K15 = 0.44 ÷ 0.64, ‫׀‬d31‫( = ׀‬0 ÷ 5) pC/N, d33 = (52 ’ 114) pC/N, ‫׀‬g31 ‫( =׀‬0 ÷ 3.2) mV∙m/N, g33 = (33 ’ 108) mV∙m/N, tgδ∙10-2 (E = 50 V/cm) = 1.0 ÷ 2.2, Qm = 8 ÷ 2000, where K15 is the electromechanical coupling factor of the shear mode of vibration. Group 6: Materials with the high stability of resonance frequency having a broad application in filter devices. Their main electrophysical characteristics are: ТC = (575 ÷ 640) K, ε33T/ε0 = 400 ÷ 1400, Kp = 0.10 ÷ 0.53, ‫׀‬d31‫( = ׀‬6 ÷ 100) pC/N, δfθ/fr ((213 ÷ 358) K) = (0.15 ÷ 0.25) %, tgδ∙10-2 (E = 50 V/cm) = 0.2 ÷ 2.0, Qm = 300÷12000 (fθ is the resonance frequency in the above temperature range, and fr is the resonance frequency at room temperature). Group 7: Materials with the low dielectric permittivity designated for the use in highfrequency acoustoelectric transducers. They possess the following electrophysical characteristics: ТC = (515 ÷ 595) K, ε33T/ε0 = 260 ÷ 510, Kp = 0.20 ÷ 0.54, ‫׀‬d31‫( = ׀‬16 ÷ 70) pC/N, ‫׀‬g31 ‫( =׀‬7.0 ’ 15.5) mV∙m/N, tgδ∙10-2 (E = 50 V/cm) = 0.3÷1.0, Qm = 200÷4500, VR = (3.60 ÷ 4.20) km/s. Group 8: Pyroelectric materials for application as operating elements of indicators of pyroelectric radiant (thermal) energy detectors for long-distance measurement of temperature of heated bodies, including moving ones. Their main electrophysical characteristics are: ε33T/ε0 = 290 ÷ 420, ‫׀‬d31‫( = ׀‬27 ÷ 49) pC/N, γ∙104 = (5.0 ÷ 5.5) C/(m2∙К), (γ∙ε33T/ε0)∙106 = (1.3 ÷ 1.7) C/(m2∙К), (γ∙d31)∙106 = (10.4 ÷ 18.5) N/(m2∙К), where γ is the pyroelectric coefficient, (γ∙ε33T/ε0) is the coefficient proportional to the V – Wt sensitivity, and (γ∙d31) is the product that characterizes the vibrational-noise stability of the piezoelectricity. Group 9: High-temperature materials that distinguish themselves by high values of the Curie point and high operating temperatures. They may be effectively used, in particular, as a base for indicators controlling processes proceeding in extreme conditions in different industrial power-plants (e.g., internal combustion engines and heated pipelines). They possess the following main electrophysical characteristics: ТC = (675 ÷ 1475) K, Тoper = (575 ÷ 1225) K, ε33T/ε0 = 48 ÷ 455, Kp = 0.015 ÷ 0.32, Kt = 0.29 ÷ 0.46, ‫׀‬d31‫( = ׀‬0.51 ÷ 35) pC/N, d33 = (12 ÷ 100) pC/N, g33 = (19.0 ÷ 33.0) mV∙m/N, tgδ∙10-2 (E=50 V/cm) = 0.1 ÷ 1.0, Qm = 100 ÷ 4000 (Kt is the electromechanical coupling factor of the thickness mode of vibration).

All of these materials are based on the PZT system and prepared using the hot-pressing method. As stated, this search for alternative materials has been of practical interest for some years because of the substantial toxicity of Pb compounds and also because of the creation of a new legislative base directed toward environmental protection. To this end, new regulations accepted by the European Union are in force as of 1 July 2006 (Directive 2002/95/EU of the

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L. A. Reznichenko, I. A. Verbenko, K. P. Andryushin et al.

European Parliament on the use of dangerous materials in electronics and electronic devices, revised on the 27th of January, 2003) [1]. This law restricts the use of lead, cadmium, mercury, hexavalent chlorine, and two-Br-substituted free radicals. Special attention is given to the elimination of Pb compounds from composition of special electrotechnical ceramics (piezoelectric ceramics). Thus, it seems advisable to proceed with only Pb-free, ecologically safe materials in electrotechnical use. Part of the materials engineering research performed at the Research Institute of SFU includes complex investigations on more than 100n-component (n = 2 ÷ 6) SS systems (more than 4 thousand of compositions) based on lead-free compounds, specifically alkali metal niobates (AMN) belonging to various structural families: perovskite (Na and K niobates), pseudoelements (Li niobates and tantalates), layered perovskite-like compounds (Sr and Ca pyroniobates), columbite (CdNb2O6, CaNb2O6, SrNb2O6), and others. A specific morphology of phase diagrams was established for binary and ternary systems, with the assistance of these compounds; which involve vast polymorphism and polymorphotropy of phase transformations. Examples of these discussed here include phase diagrams of SS of the binary ((Na, Li)NbO3, (Na, K)NbO3, (Na, Pb)(Nb, Ti)O3) and ternary ((Na, Li, K)NbO3, (Na, Li, Cd0,5)NbO3, (Na, Li, Sr0,5)NbO3) systems based on Na niobate; some peculiarities of the relationships between composition, structure, and properties were revealed; the regularities of variation of physical properties in the case of iso- and heterovalent ionic substitutions were established and a possibility of monitoring the electrophysical parameters of SS by means of variation of the conditions of formation of their structure was explored. This enabled the design of a number of novel Pb-free FPCM and the patent of the technologies used on their preparation.

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2. DIELECTRIC AND PIEZOELECTRIC PROPERTIES OF NaNbO3BASED SOLID SOLUTIONS Sodium niobate, NaNbO3, exhibits a wide range of isomorphism [2], the largest number of polymorphic transformations in the perovskite family [3], and various polar states [4, 5]. For these reasons, a wide variety of NaNbO3-based solid solutions [6], and even minor amounts of solutes may give rise to various structural instabilities [7]. These instabilities may be associated with changes in structure type, symmetry, and superstructure, or with compositional and electrical ordering, which impedes phase-diagram studies and the interpretation of results. In the piezoelectrical systems in question, the development, of longperiod (modulated) structures [8] and clustering at numerous, often overlapping morphotropic phase boundaries [9-11], leads to the formation of phases with complex mesoscopic patterns. The structure and physical characteristics of this constituents and the interaction between them determine the macroscopic properties of NaNbO3-based solid solutions. NaNbO3-containing systems have not yet been studied in sufficient detail [12, 13]. Given that such systems are of considerable scientific interest (especially with respect to the development of the theory of consecutive phase transitions in complex, spatially nonuniform condensed media) and practical importance (in the engineering of electrically active materials with unique properties [14, 15]), and also that recent findings [4, 5] have radically changed the picture of sodium niobate phase relations [4,5], a more in-depth study required of the

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phase diagrams of the systems in question and the properties of NaNbO3 based solid solutions, which form the basis of a large class of ferroelectric (FE) materials [14]. Thus we undertook detailed x-ray diffraction (XRD) and electrical studies of two NaNbO3-based solid solution systems - Na1-xLixNbO3 (0 ≤ x ≤ 0.145) (1), Na1-xKxNbO3 (0 ≤ x ≤ 1.0) (2), which also were investigated earlier [7, 15-18]. To obtain reliable, systematic data, we studied more than 50 compositions in each system, with ∆x of 0.0015 ÷ 0.0025. For each composition, eight to ten samples (10 mm in diameter and 1 mm in thickness) were examined. The samples were prepared by solid-state reactions at 1070 ÷ 1120 K over a period of 5 h, followed by hot pressing [19] at 19.6 MPa for 40 min at temperatures from 1320K to 1470 K, depending on composition. The starting materials used were reagent-grade (Li, Na, K)2CO3, extra-pure-grade TiO2, and NbO-Pt Nb2O5 (or extrapure-grade Nb2O5). The intermediate and final reaction products were examined by XRD on a DRON-3 powder diffractometer (Mn-filtered FeKα radiation). Dielectric and piezoelectric characteristics were measured at room temperature by a standard procedure [20]. Figure 1a shows the refined phase diagram of system Na1-xLixNbO3 (1). At room temperature, there are 13 different phase states within the solid-solution range ( 14.5 mol % LiNbO3), including 7 two-phase regions. Using XRD, we located three morphotropic regions corresponding to different changes in cell symmetry and regions exhibiting changes in the structure type of the solid solution. Dielectric measurements revealed regions of the antiferroelectric (AFE)-FE phase transition and a region of unusual dielectric behavior. In the composition range 0 ≤ x ≤ 0.0375, the solid solution has an orthorhombic structure analogous to that of pure NaNbO3 (O = OII in [19]), with a monoclinic (M) perovskite subcell (M4) (Figure l a, regions I-V; phase O in Megaw's notation [3]). The lattice parameters of the O cell (A, B, C) are related to those of the reduced M subcell (a0 = c0, b0, β ≠ 90°) by A = 2a0 cosβ/2, B = 4/b0, and C = 2a0 sinβ/2. In regions V-X (0.0320 ≤ x ≤ 0.118), NaNbO3 has an O(M2) cell with B = 2b0 (phase Q in [21, 22]). The first morphotropic region MR, (region V) lies in the range 0.0320 < x < 0.0375 and contains O(M4) and O(M2) phases. In the range 0.1075 ≤ x ≤ 0.1310, the unit cell is rhombohedral (R), with A = B = C = 2a0 (Figure 1a, regions X-XII; phase N in [3]). This is evidenced by the following: these solid solutions have the same structure (perovskite) as NaNbO3; in both cases, there is a superstructure with doubled lattice parameters (in NaNbO3, below 170 K) owing to tilted oxygen octahedra; the room-temperature lattice parameters of the rhombohedral solid solutions with 0.1075 ≤ х ≤ 0.1310 (a = 3.902 Ǻ, α = 89°6') differ little from those of NaNbO3 below 170 K (a = 3.908 Ǻ, α = 89°13'). The single-phase region of R is 0.1180 ≤ x ≤ 0.1250 (region XI). In the composition ranges 0.1075 < x < 0.1180 and 0.1250 < x < 0.1310, R coexists with O(M2) (Figure la, regions X and XII, or MR2 and MR3). Region XIII (0.1310 < x ≤ 0.1450) contains orthorhombic solid solutions O(M2). Figure 1b shows the composition dependences of lattice parameters for solid solutions. The nonmonotonic behavior of the lattice parameters is due to the large number of phase transitions.

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Figure 1 (a). Refined phase diagram of system Na1-xLixNbO3

Figure 1 (b).

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Figure 1 (c).

Figure 1 (d).

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Figure 1 (e). Figure 1. (a) Partial phase diagram of the NaNbO3-LiNbO3 system and the composition dependences of the measured and calculated perovskite cell parameters (b). Composition dependences of dielectric (c), piezoelectric (d) and elastic (e) properties for Na1-xLixNbO3 solid solutions. (1,2,3,4 -  calc calculated by the formulas 1,2,3,4 respectively; 5 - β; 6- α; 7- b0; 8- aR; 9 - a0 = c0; 10 - ε/ε0; 11 - ε33T/ε0; 12 - tanδ; 13 - Kp;14 - Kt; 15 - d31; 16 - d33; 17 - Qm; 18 - VR; 19 - δP; 20 - Y11E).

To identify the nature of the solid solutions, we analyzed the composition (x)

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dependence of the experimentally determined average lattice parameter V: a meas = 3 V (where V is the volume of the perovskite cell) in comparison with that of the lattice parameter acalc calculated by the formula derived for (1-x)ABO3-xA'BO3 systems [15],

acalc 

2 1  x n A LAO  xn A LAO   2nB LBO , (1  x)n A  xn A  nB

(1)

where nA, nA΄, and nB are the valences of cations A, A', and B, respectively; and LA-O, LA’-O, and LB-O are the unstrained A-O, A'-O, and B-O bond distances (CN = 6 for B and 12 for A and A') [19]. In Fig. la, line 1 represents the acalc (x) data obtained by this formula. Note that acalc differs drastically from a meas . The origin of this discrepancy is discussed below. A characteristic feature of octahedral oxides having ReO2-type [24] or perovskite [25] structures and containing ions of variable valence is that oxygen vacancies and, in mixed oxides, an equivalent number of cubo-octahedral sites can be eliminated by crystallographic shear [25]. In niobates (i.e., Nb2O5, NaNbO3, its solid solutions, and other ternary oxides [8]), this process leads to the formation of ordered blocks (Magneli phases) separated by channels with vacant or partially filled octahedral or tetrahedral interstices [8,24]. The existence of the latter in NaNbO3 and its solid solutions was proved by x-ray fluorescence spectroscopy [8]. Conclusive evidence for the presence of octahedral interstices in NaNbO3 is lacking. However, by analogy with earlier results [24], it seems likely that both defect species are present in NaNbO3. The AO-site deficiency (y), which leads to deviations from stoichiometry,

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can be evaluated using the formula for the average lattice parameter of ABO3 perovskites [23],

acalc 

n A a A  nB a B , n A  nB

where a A  2LAO and a B  2 LB O . Multiplying the first term in the numerator by 1 y and equating the resultant expression to the measured lattice parameter of NaNbO3,

ameas 

(1  y) 2 LNa O  5  2  LNb O 6

one can find y. Clearly, the deviation from stoichiometry is determined by both a meas and the ionic radii used in calculations. The present and earlier data for NaNbO3 demonstrate that the a meas of ceramics and crystals varies from 3.900 to 3.909 Ǻ, depending on the starting-mixture composition and preparation procedure. Using the LNa-O and LNb-O calculated as the sums of the corresponding ionic radii [26], we find for the above values of a meas that y varies from 0.0793 to 0.0639, respectively, and is close to the Li deficiency in the congruently melting composition of LiNbO3 [27]. With LNb-O = 2.01 Ǻ [23], we obtain that y varies from 0.0505 to 0.0351. 12 CN 6 a meas = 3.9035 Ǻ (the In this work, we used LCN Na O = 2.4576 Ǻ, LNa O = 2.01 Ǻ, and

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most typical value) and obtained y = 0.044. Thus, the AO-site deficiency in NaNbO3 is  4.4 at. %. Depending on the thermal history of the solid solution, the excess Na may be incorporated in impurity phases (in particular, those of eutectic origin, such as NaNb3O8, Na2Nb4O11, and NaNb2O5F), may reside in intergranular layers, or may dissolve in the parent structure, occupying irregular sites [8] and forming an autoisomorphous substance (or internal [28] multiplicative [29] interstitial solid solution) of composition NaNbO3. The formula of 12 CN 6( 4) this interstitial solid solution can be represented as Na CN ]NbO3 (a), where 0.956 [ Na 0.044

CN = 12 refers to Na in regular, cuboctahedral sites, and CN = 6 or 4 refers to Na in irregular, octahedral, or tetrahedral sites. Solid solution (a) contains Na+ ions in different crystalchemical environments (different oxygen coordinations). As demonstrated previously [8], by optimizing hot-pressing conditions (temperature, atmosphere, duration, pressure, and others), one can obtain essentially pore-free solid solutions containing excess Na in irregular sites, which will be considered below. Taking into account that Li0.94NbO3, thermodynamically the most stable composition of lithium niobate [27], is close in nonstoichiometry to NaNbO3 and that the LiNbO3 content of the materials studied here is not very high ( 14.5 mol %), we believe that the AO-site deficiency revealed in NaNbO3 remains unchanged over the entire composition range studied. This is confirmed by the large separation between line 1 and the a meas (x) curve at all values of x (Figure lb). With allowance made for the AO-site deficiency in Na1-xLixNbO3, the formula for acalc takes the form

ameas 

0.956nNa 2[(1  x) LNa O  xLLi O ]  nNb 2 LNb O (2) 6

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The corresponding line (Figure lb, line 2) lies well below line 1 and must describe the variation of the lattice parameter with x for substitutional solid solutions. However, it deviates markedly from the a meas (x) curve, which lies markedly lower than line 2 for x ≤ 0.0375 (Figure lb) and can be represented by a set of straight lines gradually approaching line 2. In the composition range 0.0375 < x < 0.0525, the experimental data scatter widely. At 0.0525 < x ≤ 0.10, the a meas (x) curve is very close to line 2. For x > 0.10, there are two curves which also differ from line 2. The discrepancy between line 2 and a meas (x) may arise from the presence of ions in irregular sites, which is not taken into account in formula (2) (in the range 0.0525 ≤ x ≤ 0.10, their contribution is generally not very large, as evidenced by the very small difference between acalc and a meas ), and also from the formation of different types of solid solutions. Most likely, substitutional solid solutions with a constant AO-site deficiency (SSS1) exist in the range 0.0525 ≤ x ≤ 0.10, where the data points fall close to line 2. For x ≤ 0.0375, the likely process is the formation of interstitial solid solutions (ISS'), with Na+ and Li+ ions in interstitial positions. In the range 0.0375< x < 0.0525 (region VI), where the a meas data scatter slightly, these solid solutions coexist (MR4). Figure la shows that, in regions X-XII, the a meas of the M phase is much smaller and that of the R phase is much larger than

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acalc . It seems likely that in the former (SSS1‘) and latter (SSS1‘‘) solid solutions, the AO-site deficiency is, respectively, higher and lower than that in SSS1. SSS1‘ and SSS1‘‘ exist in the composition ranges 0.095 ≤ x ≤ 0.145 and 0.1075 ≤ x ≤ 0.130, respectively. SSS1 and SSS1‘ coexist in the range 0.095 < x < 0.10 (MR6), and SSS1‘ and SSS1‘‘ coexist at 0.1075 < x < 0.130 (MR7). The distribution of ions in interstitial solid solutions also depends on composition. For x ≤ 0.044, all of the cuboctahedral sites are occupied by Na+, and the excess Na+ and incorporated Li+ ions occupy irregular, tetrahedral and octahedral sites. This is possible because Na+ may have a reduced coordination number [30] and because Li+ typically has coordination numbers other than 12 [19] and readily intercalates into various structures owing to its small size and high mobility [24]. With increasing x, the amount of Na+ in such sites decreases, and that of Li+ increases; at x  0.044, most of the irregular sites are occupied by Li+. At x ≤ 0.044, the crystal-chemical formula of the solid solution is 12 CN 6( 4) Na CN NbO3 (b). Since the actual distribution of Na+ and Li+ over 0.956 [ Na 0.044 x Li x ]

the irregular sites is unknown, as is the coefficient taking into account the contribution of the unstrained LA-O bond distances for fourfold-coordinated ions in the formula for the acalc of solid solutions (b), this formula can be written for the specially synthesized composition Na1z NbO3z / 2 (z = 0.044), containing no excess Na+, which was doped with 1 mol %

LiNbO3 in excess of stoichiometry. Assuming that all of the incorporated Li+ ions occupy octahedral sites, we obtain

acalc 

12 CN 6 CN 6 0.956nNa 2 LCN Na O  xn Li 2 LLi O  nNb 2 LNb O 6

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The corresponding line (Figure lb, line 3) deviates drastically from a meas . To examine the possibility that Li+ ions occupy tetrahedral sites, we take into account, with some 4 coefficient (α), the contribution of the unstrained LCN Nb O bond distances to the lattice

parameter:

acalc 

12 CN 4 CN 6 0.956nNa 2 LCN Na O  xn LiLLi O  nNb 2 LNb O 6

(4)

The value of α can be found by replacing acalc with the known a meas (e.g., a meas = 3.9016 Ǻ at x = 0.005). The sought value of α is 1.2599 (or  3 2 ). With this α, acalc (x) for solid solutions (b) (line 4) coincides with a meas (x) at x ≤ 0.0075. This indicates that the Li+ ions in tetrahedral sites play a key role in determining the a meas of the solid solutions under consideration, leading to a sharp decrease in a meas . At the same time, the excess Na+ is likely to occupy the octahedral and tetrahedral sites in such a way that their contributions to a meas (x) balance one other. This appears quite plausible because calculations demonstrate that, at the nonstoichiometry in question, the excess Na+ (0.044  0.0075 = 0.0365) is distributed over tetrahedral and octahedral sites in the ratio 2 : 1. In the range 0.0075 < x < 0.0375, both tetrahedral and octahedral sites are occupied. Since this increases acalc (line 3),

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a meas shows a jump-like variation in this composition range (which points to a complex redistribution of Na+ and Li+ over irregular sites), approaching the value corresponding to SSS1. At x > 0.044, there is no excess Na+ and a progressively larger amount of cuboctahedral sites are unoccupied. The number of Li+ ions per formula unit exceeds that of unoccupied A sites by 0.044, and they are distributed as follows: x  0.044 ions in cuboctahedral sites (substitutional solid solutions) and 0.044 ions in tetrahedral and octahedral sites. This relationship remains unchanged in the x range as long as nonstoichiometry y is constant (up to x = 0.10). Thus, the actual SSS1 is a combination of substitutional and interstitial solid solutions. Its crystal-chemical formula 12 CN 12 CN 12 CN 4 is (Na1CN ]NbO3 (c) where x1, is the Li+ concentration in ( x0.044Li x0.044 )0.956[Li 0.044 x1 Li x1

tetrahedral sites. The formula for acalc in the range 0.044 ≤ x ≤ 0.10 takes the form CN 12 CN 12 1 0.956 2 (1  ( x  0.044)nNa LNa O  ( x  0.044)nLi LLi O ) acalc    6 CN  4 CN 6 3 6  [(0.044  x1 )2nLi LCN  Li O  2 x1nLi LLi O ]  2n Nb LNb O

(5)

The multiplicativity of such solid solutions is due to the presence of Li+ in inequivalent crystallographic sites. Since the isomorphism of LiNbO3-rich solid solutions is mainly due to substitutions on regular sites and the role of interstitial incorporation is less important, solid solutions (c) are indicated in the phase diagram (Figure la) as SSS1. The nonmonotonic

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variation in the lattice parameters of the O and R phases at x ≥ 0.10 is associated with the existence of several morphotropic phase boundaries of different origins in regions VIII—XII. Using dielectric measurements, we revealed the composition range in which the type of electrical ordering changes: the solid solutions are AFE at x ≤ 0.015 and FE at x ≥ 0.0225. As seen in Figure lb, the unit cell of the O phase (AFE NaNbO3) is retained in the FE phase, which gives grounds to believe that the AFE-FE transition in the system under consideration involves an intermediate state (MR5) that combines both types of electrical ordering (region III, 0.015 < x < 0.0225): the AFE superstructure and polarization [31]. The unusual behavior of dielectric permittivity ε [32] (an increase in peak permittivity temperature with decreasing frequency, i. e., anomalous dispersion) and a series of additional relaxation peaks in ε due to structural defects (oxygen vacancies) occur in the range 0 < x ≤ 0.0075 (region I). Figure lc-1e shows the composition dependences of dielectric and piezoelectric characteristics and the uniform distortion parameter δ [19]. The electrical parameters are seen to pass through extrema near structural instabilities, which is attributable to anomalies in structural characteristics [19]. As distinct from system (1), sodium and potassium niobates form a continuous series of solid solutions [14]. Using XRD analysis and dielectric measurements, we revealed 13 different phase states, including 7 morphotropic phase boundaries between regions differing in the nature of the solid solution, electrical ordering, and superstructure. The orthorhombic symmetry with a monoclinic perovskite subcell, O(M4), characteristic of NaNbO3, persists up to x  0.02 (Figure 2a, regions I-1V, phase O). The solid solutions with 0.0025 ≤ x ≤ 0.22 have an O cell with B = 2b0 (O(M2)) (Figure 2a, regions II-VIII, phase Q). In the range 0.20 ≤ x ≤ 0.28, the solid solutions have the same symmetry, O(M2'), but with a far smaller β (Figure 2a, regions VIII-X, phase K in [33-35]). At 0.25 ≤ x ≤ 0.43, the symmetry of the unit cell is O(M1) (B = b) (regions X-XII, phase L in [33-35]). In regions XII and XIII (0.41 ≤ x ≤ 1.0), the solid solutions have an O cell with no superstructure, O(M0) (phase M in [33-35]).

Figure 2 (a) Partial phase diagram of the NaNbО3-KNbO3 system and the composition dependences of the measured

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Figure 2 (b) calculated perovskite cell parameters for NaNbО3-KNbO3 system

Figure 2 (c) Composition dependences of dielectric(c) properties for Nа1-хKхNbO3 solid solutions

Figure 2 (d). Composition dependences of piezoelectric properties for Nа1-хKхNbO3 solid solutions

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Figure 2 (e). Composition dependences of elastic properties for Nа1-хKхNbO3 solid solutions (1,2,3,4 calc calculated by the formulas 1,2,3,6 respectively; 5- b0; 6- a0 = c0; 7- β; 8 - ε33 /ε0; 9 - ε/ε0; 10 T

tanδ; 11 - Kp;12 - Kt; 13 - d31; 14 - d33; 15 - Qm; 16 - VR; 17 - δp; 18 - Y11 ). E

Thus, the composition ranges of morphotropic changes are 0.0025 < x < 0.02 (O(M4 + M2), MR1, regions II-IV), 0.20 < x < 0.22 (O(M2 + M2‘), MR2, region VIII), 0.25 < x < 0.28 (O(M2' + M1), MR3, region X), and 0.41 < x < 0.43 (O(M1 + M0), MR4, region XII). Table 1. Composition ranges of orthorhombic phases in the (Na1-xKx)NbO3 system Phase

x

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this work

Phase

x

[32-34]

O(M4)

x > 0.02

P

x > 0.02

O(M2)

0.0025 ≤ x ≤ 0.22

Q

0.0025 ≤ x ≤ 0.18

O(M2‘)

0.22 ≤ x ≤ 0.28

K

0.18 ≤ x ≤ 0.33

O(M1)

0.25 ≤ x ≤ 0.43

L

0.33 ≤ x ≤ 0.48

O(M0)

0.41 ≤ x ≤ 1.0

M

0.48 ≤ x ≤ 1.0

O(M4 + M2)

0.0025 < x < 0.02

P+Q

 0.0025 < x < 0.22

O(M2 + M2‘)

0.2 < x < 0.22

O(M2' + M1)

0.25 < x < 0.28

O(M1+ M0)

0.41 < х 0.15, the prevailing OP is p, since the rotational OPs appear, if at all, only together with it. For x < 0.15, the rotational orderings arise independently of polarization; that is, the instabilities have ― equal rights.‖ As noted above, the effect of thermodynamic history on the PT temperatures and the physical characteristics of the SSs in system II are most significant precisely in the range x < 0.15. Thus, correlating changes in the phase-state picture in system II with the enhanced effect of preparation conditions on the properties of appropriate SSs, one notes that enhancement takes place with the appearance of each new OP, which characterizes the complete set of phase states for a given x.

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Such a correlation may be explained as follows. A fabrication process implies a given sequence of external actions; hence, the correlation between the SS properties with the process conditions reflects different responses of the material to these actions. For some of the external actions, instabilities generating various OPs may respond oppositely. For example, hydrostatic pressure favors rotational PTs (i.e., raises their temperature) and suppresses ferroelectric ones [53]. Under certain conditions, this tendency may change the domains of existence of the phase states, the phase states themselves, and, hence, the entire set of the SS physical properties. Thus, two or more dissimilar structural instabilities having comparable strengths cause a strong dependence of the SS properties on the process conditions. This tendency shows up most vividly in system I, where the rotational and polarization instabilities coexist throughout the stability range of the perovskite phase. In SN, the domains of existence of the rotational and polarization orderings overlap heavily, while in perovskite-like titanates ATiO3 (A = Ba, Sr, or Ca), they do not overlap at all [39, 51]. It should be borne in mind that the firing of the ceramics and the growth of the SS crystals take place at temperatures far exceeding those of the PTs. Therefore, the thermodynamic history influences the structural instabilities indirectly. For example, in hotpressed ceramics, high residual mechanical stresses arise [54]. They have been shown [55– 57] to be the superposition of the hydrostatic pressure and the uniaxial compression along the axis of pressure application during hot pressing. Accordingly, these stresses variously influence the properties of the SN-based ferroelectric and antiferroelectric SSs. In SN, three high-temperature PTs are due to the rotation of the octahedra; and the following three PTs, to the combination of the rotation and polarization of the octahedra. In this case, two complex-ordered antiferroelectric phases (R and P) and one low temperature ferroelectric (N) phase arise. From Figure 3, it follows that the preparation conditions have a significant effect on the rotational PT temperatures. This means that the rotational instabilities are very sensitive to external actions. To realize the mechanism of this effect, it is necessary to consider the formation of the rotational instabilities and the phase states induced by them. The sequence of purely rotational PTs in NaNbO3 is the following: K K U ( Pm3m  Oh1 ) 915    T2 ( P4 / mbm  D45h ) 850    K T1 (Ccmm  D217h ) 790   S ( Pmmn  D213h )

The tetragonal phase T2 forms from the cubic phase U because of the condensation of the z component of the mode M3 (the associated ordering is 00). The phase T1 forms from T2 by the condensation of the y component of the mode R25 (0). Finally, the phase S arises from T1 through the condensation of one more component of the mode M3 (3). As has been shown [49, 50], this sequence of the orderings is favored by the specific features of the SN structure, which will be considered below. In the symmetric phase, the size of the Na+ cation (0.98 Å) is much less than that of the octahedral void where it is located (t  0.87 < 1). In addition, the charge of Na+ is small, so that the force attracting it to the void center is low. If t < 1, NbO2 planes are compressed, those of NaO are stretched. Compression causes a bend of Nb-O-Nb bonds, and stretching increases the rms displacements of the Na+ cation from the void center. Eventually, the degree of Na+ delocalization grows. The displacement also increases because, as mentioned, the

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force of Na+ attraction to the void center is low (Na+ has a small charge, +1). As the temperature is reduced, this effect is compensated for by bond bending due to the rotation of the octahedra. In the SN cubic phase, both rotational modes M3 and R25 become ― soft‖ as the temperature decreases but the former mode condenses first. Presumably, when the octahedra corresponding to M3 rotate, a local quadrupole moment arises in the void and Na+ cations delocalized from the void center interact with this moment, decreasing the energy of M3 and, thereby, favoring its condensation. The OP y appearing in the T2 →T1 transition is, to a great

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y extent, associated with the triple interaction of type M 3z R25 [52, 58], where the rotational

modes M3 and R25 are coupled through the vibrational mode X of the Na+ cations. Since the Na+ cation is relatively free to move in the void, this ― indirect‖ coupling turns out to be strong enough for y to follow z. From this reasoning, it follows that the substitution of a larger K+ cation (1.33 Å) for Na+ (0.98 Å) will decrease the temperatures of all the rotational PTs, since the ― free‖ space shrinks for both the anions (Nb-O-Nb bonds become difficult to bend) and the alkaline cations (the delocalization diminishes). In this case, t grows, the compression of the NbO2 planes decreases, and Nb-O-Nb bonds straighten out. Accordingly, the conditions for ferroelectric ordering become more favorable and, at x > 0.15, this type of ordering prevails, reconfiguring rotational distortions. Thus, system II becomes more stable and denser (t increases) as x rises. As a result, the manufacturability improves. The substitution of a smaller Li+ cation (0.68 Å) for Na+ must, as follows from thedescription above, extend the free space and raise the temperatures of all the rotational PTs. This is the case both in the crystal and the ceramic of system I obtained by the various techniques, but only for x > 0.04 ÷ 0.06. At smaller x, the situation observed experimentally is more complicated. In ceramic I [40], obtained by the conventional firing method without pressing, the temperatures of all the rotational PTs increase with x. In hot-pressed ceramic I [16], the temperatures of all the rotational PTs decrease with increasing x for 0 < x < 0.02, as in system II, and start rising at x > 0.02. The decrease in the PT temperatures can be related to the increase in the structure density (since the lattice constant decreases [16]) rather than to the regular bond bending. High temperature densification due to outer pressure can be associated with many factors. Among them are the isomorphic substitutional-to-interstitial transformation of the SSs (configurational PT) [59] (because of the high intercalation capacity of Li+ ions [24]), octahedron deformation, lithium–defect interaction, etc. However, it is difficult to draw any definite conclusions on this point without further investigation. The decrease in the S–R and R–P PT temperatures in hot-pressed ceramic I is likely to be associated with the fact that the antiferroelectric order rises together with the complex rotational order, and the suppression of the latter (because of the densification) causes the PT temperatures to diminish. With growing x (at x > 0.02), the Li→Na substitution dominates, which ― loosens‖ the structure (t drops) and raises the temperatures of the rotational PTs. In single crystals I, the rotational PT temperatures vary insignificantly at x < 0.04. Then (x > 0.04), they grow with x, the growth being faster for x > 0.06 (Figure 3). It seems that, at small x, the probabilities of substitution and interstice occupation are close to each other, while substitution dominates at x > 0.04, which increases the temperatures of the rotational PTs.

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Thus, the porosity of SN-based SSs is a primary factor responsible for the effect of the thermodynamic history (preparation conditions) on their physical properties. The porosity is associated with the misfit between the crystal-chemical parameters of the components and the perovskite structure (small tolerance factor t). This makes the structure unstable, specifically against rotational distortions. As the porosity grows (t decreases still further) when the SS composition changes, the material becomes even more sensitive to external actions and its response to them is more pronounced and diversified. The manufacturability of SN-based SSs diminishes appreciably at t < 0.90 ÷ 0.93, since rotational and polarization instabilities become comparable in strength in this interval of t. The manufacturability degradation is also typical of other oxides of the perovskite family for which t ≤ 0.93 and which exhibit rotational PTs, such as PbZrO3 [54] and Pb1–xCaxTiO3 at x > 0.4 [60]. Thus, the results reported can be considered as general.

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4. MODIFICATION OF (Na, Li)NbO3 SOLID SOLUTIONS WITH HETEROVALENT CATIONS The introduction of small (within 10 mol %) additions of various monoxides is one of the most effective approaches to modifying the physical properties of ferroelectric materials. The great potential of this approach is associated with the large number of metal monoxides capable of dissolving in mixed oxides, in particular, in FE perovskite oxides, in high concentrations; oxides that form fluxes with components of host materials (B2O3, SiO2, V2O5, and others); and additives that have combined effects on the properties of parent systems through the formation of a liquid phase and cation exchange with the host material (modifying glasses). One obvious advantage of this approach is the possibility of selectively tuning particular parameters of the material via slight compositional changes, without altering its inherent properties. The most detailed studies concentrate on modifying solid solutions based on lead zirconate titanate: over 300 oxide combinations have been used to date for this purpose [61]. The results were used to reveal general trends in the physical properties of modified PZT materials [61] and to predict the characteristics of new FE/piezoelectric ceramics [62]. In recent years, ever increasing attention has been given to environmentally safe (lead-free) FE materials based on alkali niobates. The possibility of modifying such solid solutions has been studied little, and most efforts have focused on the NaNbO3-KNbO3 system [12]. Niobates of the NaNbO3-LiNbO3 system are no doubt also of interest, since they offer a unique combination of properties: low density ( 4.5 g/cm3), high sound velocity ( 6 km/s), sufficiently good piezoelectric properties, extremely low dielectric permittivity ( 100), and a broad range of mechanical quality factors (Qm from tens to hundreds). This makes these niobates indispensable materials for microwave piezoelectric conversion [63]. The similarity between structural defects in octahedral oxides suggests that many of the general relationships inferred for PZT materials [19, 25, 54] are applicable to the A+B5+O3 (A = K, Na, Li; B = Nb) niobates. At the same time, the inherent crystal-chemical features of solid solutions containing alkali-metal cations on the A site contribute to structural disorder in the alkali niobates. In general, the following point defects are possible in the alkali niobates: vacancies, interstitials, A cations on the B-site, B cations on the A-site, and quasi-free

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electrons resulting from the process Oo  Vo  e  1/ 2O2 . The tendency of Nb to readily

change its oxidation state (Nb5+ → Nb4+) and the ordering of the resulting oxygen vacancies

((Nb15x Nb4x )2 O5x x, where is an oxygen vacancy) lead to elimination of point defects and formation of crystallographic shear planes. This outcome results, on the one hand, in a diversity of modification scenarios (a modifier cation may occupy a lattice site of the host crystal, leading to the formation of cation and oxygen vacancies (A cations produce B-site and oxygen vacancies, and B cations produce A-site and oxygen vacancies) [64]; may substitute for a host cation; may occupy interstitial sites; and may be present as an intergranular impurity); and on the other hand, in specific isomorphic substitutions in alkali niobates. These findings led us to select the NaNbO3-LiNbO3 system for modification in this study with the aim of investigating the effect of iso- and heterovalent cation substitutions on the physical properties of the alkali niobates, identifying promising approaches for creating nextgeneration FE/piezoelectric ceramic materials and extending the range of their potential application. We studied ceramic samples of the Na0.875Li0.125NbO3 solid solution containing modifying additions. This single-phase, rhombohedral solid solution is close in composition to the rhombohedral/orthorhombic (R/O) morphotropic phase boundary in the system and has extreme piezoelectric properties [65]. The modifiers introduced into the solid solution were Mg2+, Sr2+, La3+, Sb3+, Al3, Ti4+, and W6+. To this end, appropriate amounts of Mg(OH)2, SrCO3, La2O3, Sb2O3, A12O3, TiO2, and WO3 were added to starting mixtures. The modifier content varied from 0.1 ÷ 0.3 to 5 at %. The experimental conditions were adjusted so as to ensure the predominant incorporation of the modifiers into the solid solution according to different schemes: (Na, Li)1-nx Mxx(n-1)NbO3 (SS-1), (Na, Li)1-xMxNb n 1 n  1 O3 (SS-2), Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

1

(Na, Li)Nb1-xMxO

5

x

3 x

5n 2

x

5

 5  n (SS-3), x

1

6 x 5

(3)

2

(Na, Li)1-xxNb1-xMxO3 (SS-4), (Na, Li)Nb

(1) (2)

Mxx/5O3, M = W. (SS-5),

(4) (5)

where M is a modifier cation of valence n. The Mg2+, Sr2+, La3+, and Sb3+ cations were substituted on the A-site (SS-1, SS-2), so that their valence was higher than the valence of the host A cation (Na+, Li+): nMA > nA; the Al3+ and Ti4+ cations were substituted on the B-site (Nb5+): nMB < nB (SS-3); and W6+ was also substituted on the B-site: nMB > nB. The modifiers were introduced according to two schemes: heterovalent substitutions on the A (SS-1) or B (SS-4) site, leading to the formation of A-site vacancies; and heterovalent substitutions leading to the formation of B-site vacancies (SS-2 and SS-5). The modifier content was 0.3 ÷ 5 at % in solid solutions 1-3, and 0.1 ÷ 5 at % in solid solutions 4 and 5.

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Niobate ceramics were prepared from analytical- or reagent-grade alkali carbonates and oxides of other elements and NbO-Pt (for piezoelectric applications) or extrapure-grade niobium pentoxide. The starting materials were dried and passed through a sieve no. 0.2 ÷ 0.9. Niobium pentoxide was washed with water through a sieve no. 0.056 in order to loosen and remove coarse particles and second-phase inclusions. Occasionally, magnetic separation was used. Starting mixtures were prepared in a vibratory mill by dry mixing or with industrial alcohol. It is well known that the only process for preparing high-quality alkali-niobate-based ceramics is solid-state synthesis followed by hot pressing [66]. The synthesis conditions were chosen based on thermogravimetric (Paulik-Paulik-Erdey thermoanalytical system) and x-ray diffraction (DRON-3 powder diffractometer, Mn-filtered FeKα radiation) data, and were corrected and optimized after structural characterization and electrical measurements. The best results were obtained in two-step syntheses: t1 = t2 = 1120 K, η1 = η 2 = 5 h. The hot-pressing conditions were optimized using shrinkage (expansion-contraction) curves and were then corrected based on microstructural analysis results. The sintering temperature tsint was varied from 1220 to 1520 K, depending on the sample composition. The sintering time and pressure were fixed at η = 40 min and p = 20 MPa, respectively. In X-ray diffraction (XRD) analysis and precision measurements of lattice parameters on the DRON-3, we used the procedure described by Fesenko [19]. The average accuracy in lattice periods (a = c, b, aR) was ± 0.5%, and that in cell angles was ± 3'. X-ray densities were determined as ρx = 1.66M/V, where M is the mass per formula unit (g) and V is the perovskite cell volume (Ǻ3). The principal structural parameter was the uniform distortion parameter δ [19]. Structural perfection was assessed from the width and shape of XRD peaks, microstrain Δd/dhkl, and the size of coherently scattering domains. Microstructures were examined on an MIM-7 optical microscope (0.17- to 0.3- μm resolutions), or if a better resolution was needed, on a Tesla BS-613 electron microscope. The dielectric, piezoelectric, and elastic properties of the solid solutions were studied at room temperature in conformity with the Special Standard OST 110444-87 [20]. We measured the relative dielectric permittivity of poled (εT33/ε0) and unpoled (ε/ε0) samples, weak-field dielectric loss (tanδ), 370 K electrical resistivity (ρv), piezoelectric moduli (d33 and d3i), electromechanical coupling coefficients of radial (Kp) and thickness (Kt) modes, mechanical quality factor (Qm), sound velocity (v), and Young's modulus (E). Dielectric measurements were made at 1 kHz in the temperature range (290 ÷ 870) K using an E-7-20 LCR-meter.

4.1. Solid Solutions with nMA > nA (SS-1 And SS-2) Analysis of formulas (1) and (2) indicates that electro-neutrality of the solid solutions is maintained through the formation of A- and B-site vacancies, respectively (the light color and high resistivity, ρv ~ 1010 Ωm, of the samples suggest that the concentration of oxygen vacancies is insignificant). At equal modifier concentrations in SS-1 and SS-2, more vacancies are needed to maintain electroneutrality in SS-1 compared to SS-2. Increasing the valence of the modifier increases the vacancy concentration. Note also that A- and B-site vacancies are inequivalent because of the inequivalence of the A-O and B-O bonds (the B-O bond is more covalent [19]). Since SS-1 and SS-2 differ in vacancy concentration, they would

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be expected to differ in physical properties. Earlier results for the PZT system [25] demonstrate that solid solutions (free of impurity phases) containing only B-site vacancies cannot be prepared: vacancies are always present on both the B- and A-sites. Since PZT and (Na, Li)NbO3 are similar in crystal chemistry, it is reasonable to assume that the same refers to the latter system, and that the actual concentration of B-site vacancies in SS-2 is lower than predicted by formula (2). The SS-1 samples were prepared in the form of dense ceramics. According to XRD results, the samples consisted of a mixture of the O- and R-phases, indicating that the modifiers shifted the morphotropic phase boundary in the (Na, Li)NbO3 system to the Ophase region. The average grain size of SS-1, D = (2 ’ 6) μm, is smaller than that of the

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unmodified material, D = 18 μm (Figure 5a). Low concentrations of modifiers (Mg2+, Sr2+, La3+, and Sb3+) reduce either the spontaneous strain or the uniform distortion parameter δ, in both the orthorhombic (δO) and rhombohedral (δR) phases. As the modifier concentration is raised, δ returns to its initial level (except in the case of Mg2+). The reduction in resistivity pv upon the introduction of donor modifiers confirms that the (Na, Li)NbO3 solid solutions are n-type [64]. The SS-1 samples show increased εT33/ε0 values, mechanical losses (1/Qm), and tanδ (Figure 5). According to their effect on these properties, Mg2+, Sr2+, La3+, and Sb3+ can be classed with soft FE modifiers [67]. The shift of the morphotropic phase boundary reduces Kp (Figure 5) because, in the (Na, Li)NbO3 system, it reaches a maximum at the R-phase boundary. The effect of Sr2+, La3+, and Sb3+ on Qm and TC correlates with that on the spontaneous strain: with decreasing δR, both Qm and TC decrease. The minimum in TC at 0.3 at % La3+ and 1 at % Sb3+ correlates with the minimum in δ. Increasing the Mg2+ content from 0.5 to 5 at % increases Qm and TC, even though the spontaneous strain of the R and O cells decreases.

Figure 5 (a).

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Figure 5 (b)

Figure 5 (c). Figure 5. Effect of the Mg, Sr+, La3+, and Sb3+ modifiers on the structural (a), dielectric (b) and elastic (c) properties of SS-1. (1 - δО; 2 - δR; 3 -

; 4 - TC; 5- ε33T/ε0; 6 - ρv; 7 - tanδ; 8 - Kp; 9 - Qm; 10 - E).

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One possible reason for this is the sharp drop in grain size, which reduces the density of domain walls [68], leads to weaker interaction between the domain walls and vacancies, and increases the grain-boundary space charge [69]. At moderate contents (≤ 2.5 at %) of trivalent modifiers (La3+ and Sb3+), these charges increase εT33/ε0 more significantly than do Mg2+ and Sr2+, which is probably associated with the formation of higher vacancy concentrations. La3+ has the strongest effect on εT33/ε0 . Analysis of the properties of the SS-2 samples indicates that, like SS-1 samples, they contain no impurity phases, consist of fine grains, and have increased conductivity (Figure 6). The effect of Mg2+, La3+, and Sb3+ on the spontaneous strain, Qm, tanδ, εT33/ε0, v, and E of SS2 indicates that, in this system, these cations also act as soft FE modifiers. At a given modifier concentration, Qm decreases in the order Sb3+ > Mg2+ > La3+. The extrema in Qm and εT33/ε0 at 1 at % Mg2+, or Sb3+ and 3 at % La3+, correspond to the R-phase boundary (Figure 6). In contrast to the SS-2 samples containing Mg2+, La3+, and Sb3+, Sr2+-doped SS-2 has reduced εT33/ε0, tanδ, and 1/ Qm values and increased v and Y11E, which seems to be due to the sharp increase in grain size at low Sr2+ concentrations. Consider now the effects of soft FE modifiers that exhibit similar behaviors in the SS-1 and SS-2 systems on the relative change in mechanical Qm / Qm (where Qm is the quality 0 factor of the unmodified material). As seen in Figure 7, Qm / Qm increases with a decrease 0 in unstrained cation-oxygen (modifier-oxygen) bond length l0 and with an increase in the electronegativity of the modifier in the corresponding oxidation state. In the SS-2 system, the effect of electronegativity on Qm / Qm is stronger than in SS-1. These data correlate with the 0

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known modification results for PZTs. The correlation between the FE hardness, quantified by 2+ Qm / Qm0 , and electronegativity may be due to the following: The introduction of Mg ,

La3+, and Sb3+ ions increases the covalence of A-O bonds, making them more directional and increasing their inequivalence. This, in turn, increases atomic displacements and, hence, the spontaneous strain [19] and the associated FE hardness [70]. Since Mg2+, La3+, and Sb3+ have similar effects on the properties of the solid solutions, it is convenient to compare the properties of SS-1 and SS-2 modified with lanthanum (Figures 5, 6). At low modifier contents (≤ 1.5 at %), the SS-1 materials are softer FEs than are SS-2 according to their δR, Qm, and TC. At modifier contents above 1.5 at %, the SS-2 materials are softer. The difference in FE hardness between SS-1 and SS-2 stems from the inequivalence of vacancies in these materials: at low modifier contents, the vacancy concentration seems to be critical (the SS-1 materials, containing more vacancies, are softer FEs than are SS-2); at increased modifier contents, the position of the vacancies in the crystal structure appears more important (the SS-2 materials, containing vacancies in the more covalent B sublattice, are softer). The lower FE hardness of SS-2 seems to be responsible for the stronger effect of electronegativity on Qm / Qm in these materials: the Qm of softer FEs is easier to raise via 0

modification. Note that at high modifier concentrations, Qm / Qm does not correlate with δR, 0 Qm, or TC. We believe that this is due to the 90° domain reorientation upon poling, as supported by the fulfillment of the relation εT33/ε0 < ε/ε0 in SS-2.

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Figure 6 (a)

Figure 6 (b).

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Figure 6 (c). Figure 6. Effect of the Mg2+, Si2+, La, and Sb modifiers on the structural (a), dielectric (b) and elastic (c) ; 4 - TС; 5 - ε33T/ε0; 6 - ρv; 7 - tanδ; 8 - Kp; 9 - Qm; 10 - E; 11 - V).

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properties of SS-2. (1 - δ0; 2 - δR; 3 -

Figure 7. Plots of Qm / Qm vs electronegativity of modifiers (a) and l0 (b); the dashed lines represent 0 the data for SS-2; the numbers at the curves specify the modifier content (at %).

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Figure 8. Lattice parameters of SS-3 as functions of (a) Al3+ and (b) Ti4+ contents. (1, 7 - α; 2, 8 - β; 3, 9 - b; 4, 10 -

0;

5, 12 - aR; 6, 11 - a0 = c0).

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4.2. Solid Solutions with nMB < nB (SS-3) Figure 8 illustrates the effects of Al3+ and Ti4+ on the perovskite cell parameters of SS-3. In both instances, starting at 0.2 % modifier, the samples consist of a mixture of the O- and Rphases. In the case of Al3+, however, both phases are present over the entire range of modifier concentrations studied. In the case of Ti4+, the two phases coexist in the composition range 0.002 ≤ x ≤ 0.005; for x > 0.005, the solid solution is single-phase and has an orthorhombically distorted structure. This suggests that the modifiers under consideration shift the morphotropic phase boundary in the (Na, Li)NbO3 system to higher lithium contents. The shift is larger for Ti4+. Thus, the effect of Ti4+ on the structure of the host material is so strong that the cell symmetry depends on the modifier content: R for x < 0.002, O + R in the range 0.002 ≤ x ≤ 0.005, and O for x > 0.005. Therefore, only the (Na, Li)Nb1-(0.2-0.5)MO0.20.5O 0.2  0.5 solid solutions fall within the morphotropic region; the other Ti-containing solid 3

2

solutions are single-phase, and their compositions lie beyond the morphotropic region, on both sides of it. Ti4+ has a stronger effect on the lattice parameters of the solid solution in comparison with Al3+. In the latter case, a0  3 V remains unchanged over the entire range of modifier concentrations studied, and a = c, aR, α, and β increase slightly for x > 0.005 (Figure 8). The lattice periods of the Ti4+-containing solid solution vary most rapidly at low modifier contents (x < 0.01), with a flat maximum at x = 0.005, corresponding to the R-phase boundary. For x > 0.01, the lattice periods are very small. At the same time, the cell angles vary significantly and nonmonotonically across the entire two-phase region. At 1.5 at % Ti4+, β has a minimum (Figure 8b). The variations in δR and δO differ markedly, but both depend little on the modifier (Al3+ or Ti4+). δR drops sharply for x ≤ 0.002 and rises slightly for x > 0.003, being substantially smaller than that in the unmodified solid solution. In the composition range 0.002 < x < 0.01, δR varies anomalously and has a maximum at x = 0.005. In the range 0.01 < x < 0.015, δO decreases; at higher modifier contents, δO varies only slightly. Ti4+ has a stronger effect on δO

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Figure 9 (a)

Figure 9 (b)

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Figure 9 (c)

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Figure 9. Effects of Al and Ti contents on the structural (a) and dielectric (b) and elastic (c) properties of SS-3. (1 - δ0; 2 - δR; 3 - ρ; 4 - TC; 5 - ε33T/ε0; 6 - ρv; 7 - tanδ; 8 - tsin; 9 - Kp; 10 - Qm; 11 - E; 12 - V).

than does Al3+ (Figure 9). Note that δR drops sharply as the Al3+ content increases to 1 at % and varies little in the range 1-3 at % Al3+. It is, therefore, reasonable to assume that Al3+ occupies interstitial and vacant sites or more likely is present in the form of an intergranular liquid phase. The acceptor modifiers Al3+ and Ti4+ would be expected to raise ρv since the (Na, Li)NbO3 solid solution is known to be n-type [66]. This is, however, not the case: both modifiers reduce ρv by almost one order of magnitude in comparison with the parent solid solution. Moreover, ρv varies stepwise, with an inflection at x = 0.01. The ρv of the Ti4+-modified solid solutions is seen to vary more significantly (x < 0.015) than that of the Al3+-modified solid solutions (Figure 9), which can be understood in terms of structural disordering in the materials under consideration. Consider the possibility that lithium occupies some of the Nb5+ sites in both the unmodified solid solution and LiNbO3 [64]. Some of the B-site modifier then substitutes not for Nb but for Li, whose valence is lower than those of Al3+ and Ti4+, so that these modifiers act as donors, reducing ρv. Partial Nb substitution on the Li site would lead to the formation of B-site vacancies, which will be negatively charged relative to the crystal and will trap electrons [64]. Al3+ and Ti4+ on the B-site will reduce the concentration of B-site vacancies, thereby carrying the donor-acceptor equilibrium in the system to a higher electron

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concentration. Clearly, some of the modifier will replace niobium, but the associated changes in ρv will probably be substantially weaker than those resulting from the donor behavior of Al3+ and Ti4+. The p-type conductivity of LiNbO3 may also have a significant effect on ρv making the competing conduction mechanisms equally probable in the modified solid solution. Note that the effects of Al3+ and Ti4+ on the ρv of LiNbO3 depend on their concentrations. In addition to reducing ρv and δR, the donor properties of Al3+ and Ti4+ show up in the behavior of εT33/ε0 and TC: εT33/ε0 increases, while TC drops in comparison with the unmodified solid solution. Ti4+ has a stronger effect on the properties of the solid solution: certain Ti4+ concentrations increase εT33/ε0 by more than a factor of 2 and reduce TC by 20%, whereas Al3+ causes no systematic variations in these parameters. In both cases (Al3+ and Ti4+), the composition dependences of εT33/ε0 and TC are nonmonotonic, with extrema at 0.2 and 1.5 at. % modifier (Figure 9). According to their effects on these parameters, both modifiers can be classed with soft FEs, as in the cases of the donor modification of PZT and (Na, Li)NbO3 (SS-1 and SS-2). It follows from earlier results [69] that both modifiers must increase mechanical (1/Qm) and dielectric (tanδ) losses, as observed at Ti4+ contents above 0.5 at %. At lower modifier contents, Qm rises and tanδ drops, indicating that the modifiers act as hard FEs. Moreover, Qm and δO vary in a similar way: an increase in δO, resulting in a lower domain-wall mobility, is accompanied by an increase in Qm, and a reduction in lattice distortion, facilitating the displacement of domain walls, leads to a reduction in Qm. Given that the mobility of domain walls depends significantly on the concentration of oxygen vacancies (which are known to reduce their mobility [12]), we believe that oxygen vacancies play a key role in determining the behavior of Qm. Since the valence of the modifier influences the concentration of vacancies that compensate the charge of the modifier ions, the interaction between vacancies and domain walls influences the FE hardness of the system. With increasing modifier content, the concentration of oxygen vacancies increases, and the vacancies might be expected to interact with domain walls, reducing their mobility and, hence, raising Qm, as observed at low modifier concentrations. The further drop in Qm is attributable to vacancy ordering, which leads to the elimination of vacancies and formation of crystallographic shear planes [24], as evidenced by XRD data. This process occurs most rapidly (at lower modifier concentrations) in the Ti4+-modified solid solutions because the structural defects in TiO2 have a tendency to be ordered [24]. That vacancies have a significant effect on Qm is also evidenced by the more significant variation of this parameter in the case of Al3+, which produces more vacancies than does Ti4+. Defect ordering is confirmed, albeit indirectly, by the increase in sintering temperature tsint after a slight initial decrease (at low modifier concentrations) and the extremum in tsint near x corresponding to the onset of oxygen vacancy elimination. Since sintering occurs through substitutional diffusion [54], vacancies play an important role in this process: the increase in vacancy concentration at low modifier contents is accompanied by a drop in tsint; at higher modifier contents, the vacancy concentration decreases, and tsint rises (Figure 9).

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4.3. Solid Solutions with nMB > nB (SS-4 And SS-5)

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Figure 10 illustrates the effect of W6+ content on the perovskite cell parameters of SS-4. It can be seen that even a very small amount of W6+ (x = 0.001) leads to the formation of a two-phase material (O + R), sharply reducing the content of the R-phase, from 100% in the unmodified solid solution to 5 ÷ 15% in SS-4 and SS-5. The uniform distortion parameter of the O cell increases with x, while that of the R cell decreases.

Figure 10. Perovskite cell parameters of SS-4 as functions of W6+ content. (1 - α; 2 - β; 3 - b; 4 aR; 6 – a = c; 7 - VO; 8 - VR; 9 - μ).

;5-

The unit-cell volume of the predominant O-phase (and that of the R-phase in SS-5) decreases for x > 0.03, which can be understood in terms of a steric factor: substitution of W6+ (ionic radius of 0.60 Ǻ) for the larger sized cation Nb5+ (0.64 Ǻ). As seen in Figure 10, the lattice parameters of the solid solutions vary more rapidly at larger x. The electrical properties of the solid solutions (Figure 11) correlate with their structural properties. The R subcell has a stronger effect on the dielectric and piezoelectric properties. In accordance with well-known relations, a reduction in δR is accompanied by a rise in εT33/ε0 (εT33/ε0 ~ 1/δ) and tanδ (tanδ ~ 1/δ) and a decrease in TC (TC ~ δ). The reason for the sharp drop in Kp is that the composition of the solid solution shifts from the R-phase region, where Kp has a maximum, to the two-phase region, where all the characteristics have reduced values. E varies little, except for the minimum in the composition range 0 < x ≤ 0.01. It is of interest to note that the W6+ donor produces no significant changes in ρv except for the small minimum at low doping levels. It seems likely that the nonmonotonic variation of the parameters in question is associated with W6+ redistribution over regular and vacant lattice sites of the solid solutions.

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Figure 11 (a)

Figure 11 (b)

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Figure 11 (c).

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Figure. 11. Effect of W6+ content on the structural (a) and dielectric(b) and elastic (c) properties of SS-4 and SS-5. (1 - δO; 2 - δR; 3 - ρ; 4 - TC; 5 - ε33T/ε0; 6 - ρv; 7 - tanδ; 8 - Kp; 9 - Qm; 10 - E; 11 - V).

The present results attest to a number of general trends in the behavior of modified solid solutions. When introduced into Na0.875Li0.125NbO3, all of the modifiers except Ti shift the morphotropic phase boundary to the O-phase region, reducing δ and TC and raising εT33/ε0, l/Qm. and tanδ. The modified Na0.875Li0.125NbO3-based solid solutions differ in FE hardness, which is due to the inequivalence of the vacancies formed: at low modifier concentrations, the vacancy concentration is critical (the solid solutions with high concentrations of A-site vacancies are softer FEs); at increased modifier contents, the position of the vacancies in the crystal structure is more important than their concentration (the solid solutions containing vacancies in the more covalent B sublattice are softer FEs). The observed reduction in resistivity upon the modification of Na0.875Li0.125NbO3 with cations such that nMB ≤ nB can be understood in terms of structural disorder in (Na, Li)NbO3 (disordering of the niobium and lithium on the A- and B-sites). The point defect density in the modified solid solutions is limited by vacancy ordering, which leads to the elimination of vacancies and formation of crystallographic shear planes. The present results suggest that some of the modified solid solutions are suitable for further optimization with the aim of creating materials that would combine high values of Qm,

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Kp, TC, ρv and v with low εT33/ε0 and tanδ, and would therefore be suitable for use in highfrequency electromechanical transducers.

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5. THE PHASE DIAGRAM AND PROPERTIES OF SOLID SOLUTIONS OF THE TERNARY SODIUM-LITHIUM-POTASSIUM NIOBATE SYSTEM This work represents a continuation of the study of ternary systems of the (Na, Li, A')NbO3 type (A' = Pb0.5, Sr0.5, Cd0.5) reported in [10, 11, 71–73]. The object of investigation was the xNaNbO3–yLiNbO3–zKNbO3 system. The samples of solid solutions were prepared by two stage solid-state synthesis (1125 K, 5 h; 1175 K, 4 h). This was followed by hot molding (19.6 MPa, 40 min, 1225 ÷ 1475 K, depending on thecomposition), ensuring a sufficiently high relative densityof the samples (1 = 0.985, where  =1/ where  is the density and 2 is the X-ray density). The boundaries of single-phase regions in the(Na, Li, K)NbO3 ternary system, the kinetics and mechanismsof interaction of the initial components, and thesequence of reactions involved in the solid-state synthesisof solid solutions were previously studied byFreidenfeld et al. [74]. The system was considered asrepresenting quasibinary compositions of the type (1-y)(NaxK1–x)NbO3–yLiNbO3 (0 ≤ y ≤ 0.20, 0.45 ≤ x ≤ 0.50). The content of NaNbO3 was varied within the limits providing for the best piezoelectric properties [12]. We studied solid solutions belonging to six z-sections corresponding to the compositions with 2.5 ÷ 15.0 mol % KNbO3. Each of these sections was represented by a series of samples containing 1.0 ÷ 15.0 mol % LiNbO3. The upper limit of these y-sections was selected so as to restrict the series of continuous solid solutions in the (Na, Li)NbO3 binary system to 14.5 mol % LiNbO3 [16]. The upper limit of the z-sections was determined by the condition that a single-phase orthorhombic structure (R = RII) [19]), analogous to that realized in the (Na, Li)NbO3 system with 0.032x0.118 [59] and complicated by the phase transitions known in the (Na, K)NbO3 system with x > 0.15 [12], would exist almost in the entire range of the component concentrations studied (except for a very narrow region close to NaNbO3). This orthorhombic structure is characterized by a monoclinic (M2) perovskite subcell with the parameters a0 = c0, b0, and 90related to those of the unit R-cell by the relationships A = 2a0cos/2, B = 2b0, and C = 2a0sin. It was of interest to study (as was done previously [10, 11, 71–73]) only a part of the phase diagram in the NaNbO3 region, where the solid solutions retain certain special properties [75] related to the proximity to this very composition. The phase diagram of the system studied (Figure 12) was determined by the diagrams of the component binary systems (Na, Li)NbO3 and (Na, K)NbO3 for the corresponding component concentrations. The NaNbO3 region is occupied by the R(M4) phase (with B = 4b0), which is characteristic of this compound at room temperature [76] and transforms in both systems into the R(M2) phase through a morphotropic region containing R-structures of various multiplicity [3]. As the z value increases, the field occupied by the R(M2) phase exhibits sharp narrowing. At the same time, the boundaries of coexistence of the orthorhombic R(M2) and rhombohedral (Rh) phases expand to form a region with a maximum width of ~10 mol % (with respect to y), which is much greater as compared to ~ 1 mol % in

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the (Na, Li)NbO3 system [16]. The Rh phase region also significantly expands. The righthand boundary of the adjacent morphotropic region featuring the coexisting Rh + R(M2) phases is conditionally indicated by a dashed line, since a more precise determination in this narrow concentration range is difficult. The width of the next R(M2) phase region is virtually the same for all z. An increase in the LiNbO3 concentration above ÷ mol % (for various z) leads to the formation of an insignificant amount (1.0 ÷ 8.0 mol %) of the impurity phase: LiNbO3 for z < 2.5 mol % and (Na, K)Nb3O8 for z > 2.5 mol % (the greater the KNbO3 content in the system, the earlier the onset of the impurity formation). This result is consistent with the data [74] concerning a decrease in the solubility of LiNbO3 in (Na, K)NbO3 and is indicative of limitations in the solid-state synthesis of solid solutions in the system with sufficiently large KNbO3 content

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Figure 12. The NaNbO3 corner of a phase diagram of the (Na, Li, K)NbO3 ternary system.

Note that, since no Rh phases appear in the (Na, K)NbO3 system [12], the regions of R(M2) + Rh, Rh, and Rh + R(M2) will probably be closed for large KNbO3 concentrations. Figure 13 shows the plots of various physical characteristics versus composition, including data for the relative permittivity of samples before  and after  polarization, planar electromechanical coupling coefficient (Kp), piezoelectric modulus (d31), mechanical figure of merit (Qm), sound velocity (VR), Young‘s modulus, and sintering temperature (t) for the solid solutions with z = 2.5 ÷ 10 mol %. Table 5 gives analogous characteristics for the solid solutions with z = 12.5 ÷ 15.0 mol %. The observed nonmonotonic behavior is obviously related to the complicated shape of the phase diagram of this ternary system, featuring a large number of interphase boundaries. Inside the morphotropic region, this behavior is probably related to a change in the phase relationships involved in the composition variations accompanied by a nonmonotonic change in the structural parameters. Table 6 indicates the main characteristics of some solid solutions of the (Na, Li, K)NbO3 ternary system promising from the standpoint of practical (piezoelectric) applications. Compositions 1 and 2 are characterized by a very low value of and  very high VR in combination with sufficiently large (for these ) Kp and d31 and moderate Qm. The VR

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and  values are advantageous for a high-frequency operation range of piezotransducers based on these compositions.

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Figure 13. The plots of some physical characteristics versus composition (LiNbO3 content y) for solid solutions of the (Na, Li, K)NbO3 ternary system in the sections with z = 2.5–10.0 mol %. (1, 9 - ε/ε0; 2, 10 - ε33T/ε0; 3, 11 - Kp; 4, 12 - d31; 5, 13 - Qm; 6, 14 - VR; 7, 15 - Y11; 8, 16 - T).

Figure 13. Contd. (17, 25 - ε/ε0; 18, 26 - ε33T/ε0; 19, 27 - Kp; 20, 28 - d31; 21, 29 - Qm; 22, 30 - VR; 23, 31 - Y11; 24, 32 – T).

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In addition, the large VR value allows a preset frequency to be obtained using thinner piezoelectric plates, which simplifies the manufacturing technology (by increasing the resonance size of devices).

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Table 5. Compositions, physical characteristics, and sintering temperatures of the (Na, Li, K)NbO3 solid solutions with z = 12.5 and 15 mol % Composition

t, K

ρ, g/cm3

TC, K

ε/ε0

εT33/ε0

K0.125Na0.865Li0.01NbO3 K0.125Na0.850Li0.025NbO3 K0.125Na0.835Li0.040NbO3 K0.125Na0.815Li0.06NbO3 K0.125Na0.785Li0.09NbO3 K0.125Na0.7875Li0.0875NbO3 K0.125Na0.7788Li0.0962NbO3 K0.125Na0.7656Li0.1094NbO3 K0.125Na0.7613Li0.1137NbO3 K0.125Na0.7545Li0.1205NbO3 K0.150Na0.800Li0.05NbO3 K0.150Na0.780Li0.07NbO3 K0.150Na0.7650Li0.085NbO3 K0.150Na0.7665Li0.0935NbO3 K0.150Na0.7438Li0.1062NbO3 K0.150Na0.7395Li0.1105NbO3 K0.150Na0.7395Li0.1105NbO3

1398 1313 1298 1303 1373 1373 1353 1273 1273 1223 1418 1363 1388 1353 1373 1273 1253

4.346 4.409 4.397 4.368 4.425 4.475 4.54 4.319 4.320 4.410 4.357 4.486 4.333 4.500 4.451 4.327 4.452

553 573 520 537 536 558 563 532 553 548 608 563 505 553 546 547 548

299 272 273 245 210 236 219 184 740 1070 250 258 223 184 353 269 650

179 165 168 173 141 170 169 145 710 980 190 168 154 133 258 143 420

Composition K0.125Na0.865Li0.01NbO3 K0.125Na0.850Li0.025NbO3 K0.125Na0.835Li0.040NbO3 K0.125Na0.815Li0.06NbO3 K0.125Na0.785Li0.09NbO3 K0.125Na0.7875Li0.0875NbO3 K0.125Na0.7788Li0.0962NbO3 K0.125Na0.7656Li0.1094NbO3 K0.125Na0.7613Li0.1137NbO3 K0.125Na0.7545Li0.1205NbO3 K0.150Na0.800Li0.05NbO3 K0.150Na0.780Li0.07NbO3 K0.150Na0.7650Li0.085NbO3 K0.150Na0.7665Li0.0935NbO3 K0.150Na0.7438Li0.1062NbO3 K0.150Na0.7395Li0.1105NbO3 K0.150Na0.7395Li0.1105NbO3

Kp |d31|, pC/N Qm VR, km/s 0,232 15.4 242 5.73 0.229 14.6 303 5.71 0.229 15.2 248 5.50 0.224 15.3 146 5.39 0.274 16.2 205 5.50 0.186 11.8 409 5.68 0.134 9.0 251 5.31 0.206 15.2 67 4.67 0.254 17.4 164 5.06 0.206 18.0 253 5.18 0.170 11.4 235 5.72 0.226 14.2 183 5.65 0.268 16.9 222 5.47 0.236 16.0 190 4.66 0.129 10.1 447 5.68 0.202 13.7 141 4.82 0.240 14.2 170 5.34 Notes: TC is the Curie temperature; v is the bulk resistivity (at 373 K).

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ρv, Ωm (373 K) 1.5 × 1010 4.2 × 108 0.7 × 1010 2.7 × 109 2.6 × 109 1.0 × 109 0.6 × 1010 2.8 × 109 2.5 × 109 0.4 × 1010 1.5 × 109 1.3 × 109 0.8 × 1010 2.5 × 109 1.6 × 1010 9.1 × 108 0.4 × 1010

YE11 ×10-11 N/m2 1.36 1.37 1.26 1.19 1.22 1.34 1.19 0.90 1.0 1.11 1.33 1.35 1.19 0.90 1.33 0.92 1.15

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Table 6. Compositions, physical characteristics, sintering temperatures, and densities of (Na, Li, K)NbO3 solid solutions promising for the piezoelectric applications Composition K0.025Na0.965Li0.07NbO3 K0.025Na0.885Li0.09NbO3 K0.05Na0.8625Li0.1235NbO3 K0.075Na0.8047Li0.1202NbO3 K0.075Na0.8094Li0.1156NbO3 Composition K0.025Na0.965Li0.07NbO3 K0.025Na0.885Li0.09NbO3 K0.05Na0.8625Li0.1235NbO3 K0.075Na0.8047Li0.1202NbO3 K0.075Na0.8094Li0.1156NbO3

T, K 1333 1373 1273 1313 1313 Kp (Kt) 0.214 0.210 0.135 0.266 0.055 (0.352)

|d31|, pC/N (|d33|, pC/N) 11.7 11.3 20.7 39.5 3.3 (21.1)

εT33/ε0 126 101 906 725 161

TC, K 613 653 520 560 577

ρ, g/cm3 4.416 4.243 4.435 4.477 4.566

Qm

VR, km/s

YE11 ×10-11, N/m2

473 121 77 77 1093

5.28 5.25 5.51 5.02 5.71

0.88 1.07 1.23 1.06 1.4

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Note: Kt is the electromechanical coupling coefficient of the transverse oscillation mode.

Another advantage is a decrease in the transducer capacitance. Finally, a low  value facilitates matching of the transducer to both generator and load. The Kp and d31 values provide for a sufficiently high efficiency of the piezotransducer, while the moderate Qm accounts for a sufficiently uniform amplitude frequency characteristic and allows the use of short pulses. A low material density leads to a significant decrease in the device weight (a decisive factor in applications such as the aerospace technologies) and in the acoustic impedance (which is also important for proper matching to the acoustic load). The high Curie temperatures Tc of these materials increase the working temperature range of the transducers. Compositions 3 and 4 (Table 2) possessing moderate  values in combination with high d31 and VR can be used as a base for the active piezoelectric materials operating in the medium frequency range. The high VR values allow medium-frequency piezotransducers to be obtained that are capable of exciting metal resonators with high sound velocities. Composition 5, which possesses a low  at a high Qm, exhibits increased anisotropy of the piezoelectric properties (Kt/Kp and d33/d31 6), which favors suppression of the spurious oscillations. Such a material can be employed in the transducers for ultrasonic defect detectors, acceleration sensors, thickness meters, and in the high-frequency instrumentation for nondestructive material monitoring and medical diagnostics.

6. PHASE DIAGRAMS AND FERROELECTRIC PROPERTIES OF SOLID SOLUTIONS OF THE TERNARY SYSTEMS (Na, Li, Cd0.5)NbO3 AND (Na, Li, Sr0.5)NbO3 The present research is a continuation and a refinement of the works [71, 77]. Based on some additional structural studies, more precise phase diagrams of the systems (Na, Li,

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Cd0.5)NbO3 and (Na, Li, Sr0.5)NbO3 were constructed that allowed a more detailed interpretation of their physical properties. As in [71,77], the systems were studied by means of z-sections that corresponded to 5 ÷ 20 mol.% Cd0.5NbO3 and 2 ÷ 50 mol.% Sr0.5NbO3 as well as y-sections with the LiNbO3 content of 1 ÷ 2 mol.% to 15 mol.%. The regimes of synthesis and sintering of the samples are given in [71, 77].

6.1. Solid Solutions of the (Na, Li, Cd0.5)NbO3 System

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Figure 14 shows a portion of the phase diagram adjacent to NaNbO3. The thin lines show y- and z-sections and the thicker ones indicate the boundary lines of the regions of different symmetry (one-, two- and three- phase ones). The phase diagram of the ternary system agrees with the diagrams of the binary systems limiting it [7, 78, 79], and, accordingly, we see wide regions of crystallization of the NaNbO3-based solid solutions (SS) of orthorhombic structure with quadruplication (phase M4) and reduplication (phase M2) along the b axis, the tetragonal structure (phase T4) and the narrow regions of rhombohedral (Rh) and orthorhombic (M2) structures together with the regions of their coexistence.

Figure 14. Phase diagram of the ternary system (Na, Li, Cd0.5)NbO3.

As is well-known [14, 62, 72], the electrophysical parameters of ferroelectric SS of different systems containing morphotropic regions (MR) have extreme values in the vicinity of MR which correlate with the values of structural parameters, in particular, with the homogeneous deformation parameter δ [80]. Let us consider the best studied sections of the system which, in addition, pass through the largest number of phases and MR. Among the z-sections, satisfying these conditions is the section z = 5 mol. % which passes through three phases M4, M2, Rh and three MR (a narrow two-phase MR1 (M4+M2), a wide three-phase MR2 (M2+T4+Rh) and a very narrow two-phase MR3 (Rh+M2)) (Figure 15). Figure 15 shows the concentration dependences of the parameter δ, the dielectric permittivities ε/ε0 and ε33T/ε0 and the piezoelectric parameters Kp, d31, g31, the

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dielectric loss tangent tgδ, the mechanical quality factor Qm, the sound speed VR and Young's modulus YE11. It may be seen that both the dielectric permittivities of SS and the parameters Kp and d31 have two maxima: the larger maxima are placed near the center of the three-phase MR2 and the smaller ones are at the right-hand boundary of the narrow MR1. In addition, all these parameters pass through the minima inside the phase M2. As to the piezoelectric parameter g31 which is known to be proportional to the remanent polarization Pr, its maxima are shifted to the left from both MR1 and MR2 to the side of the phases M2 and M4 (in the phase M4 this maximum is not fully examined because of the absence of corresponding samples). Such positions of the maxima of g31 are observed usually in ferroelectric systems [14, 62]. The considered dependences of the electrophysical parameters may be related to the behavior of the parameter δ, which is minimum inside both MRs and passes through the maximum in the phase M2. (The extreme narrowness of MO3 makes it difficult to take into account its influence on the concentration dependences of the parameters). Of special interest is the fact that in Figure 2a the larger maximum of ε33T/ε0 is located not outside the right-hand boundary of the wide MR2 [14,62] but inside MR2. The same situation was observed in some sections of the (Na, Li, Pb0.5)NbO3 system [72] and this was explained, in particular, by coexistence of two phases (M and Rh) leading to a considerable increase in the number of possible directions of the spontaneous polarization vector N = 8(Rh)+12(M) =20 as compared with the pure phases. (In PZT-based systems [14,62] this number is much smaller: N = 8(Rh)+6(T) =14 and does not, practically, affect the position of the ε33T/ε0 maximum).

Figure 15 (a)

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Figure 15 (b)

Figure 15 (c). Figure 15. Dependences of dielectric (a), piezoelectric (b), structural and elastic (c) characteristics of SS of the (Na, Li, Cd0.5)NbO3 system on the LiNbO3 content for the section z = 5 mol.% Cd0.5NbO3. (1 ε/ε0; 2 - ε33T/ε0; 3 - tanδ; 4 - Kp; 5 - d31; 6 - g31; 7 - Qm; 8 - VR; 9 - δRh; 10 - δT; 11 - δM2; 12 - Y11E).

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In the case under consideration, in the three-phase MR2, N is much larger: N = 8(Rh)+12(M)+6(T) = 26 which leads to a remarkable increase of the orientation part of dielectric permittivity inside MR and determines the corresponding position of maxima of both ε/ε0 and ε33T/ε0. This, in turn, has an influence on the positions of the Kp and d31 maxima, which are only slightly affected by the magnitude of g31 which anyway is small in this system. A similar situation is observed in the narrow MR1, where N = 12(M2)+12(M4) = 24 and the ε33T/ε0 maximum has a larger effect on the Kp and d31 values than the g31 maximum. Concentration dependences of the parameters tgδ, Qm, VR and YE11 can be explained in terms of ferrostiffness of SS characterizing stability of the domain structure against external influences [62]. As shown in [62], with the enhancement of ferrostiffness the parameters δ, Qm. VR, YE11 increase while ε33T/ε0 and tanδ decrease. This leads to the closeness of positions of the ε33T/ε0 and tanδ maxima and minima, whereas the positions of the Qm, VR and YE11 maxima are close to the position of the ε33T/ε0 minimum (and vice versa). Among the y-sections the most clear-cut regularities in variations of the parameters can be observed at y = 4 mol.%. This section passes through the regions of the phases M2 and T4 and the two MRs: a wide MR1(T4+M2) and a narrow MR2 (M4+M2) (see Figure 14). The regularities in the variation of electrophysical parameters are similar to those observed in most ferroelectric systems [14, 62] and correlate with the behavior of δ. As in the majority of the niobate systems, compositions of the system under consideration admit a high sound speed and have a low density and also a high Curie temperature TC ( > 670 K) as well as a wide spectrum of dielectric permittivity values (from 160 to 1000 ÷ 1200) at the acceptable piezoelectric parameters. This enables one to use them in high-temperature transducers operating in high-and midfrequency ranges. Table 7 presents some compositions possessing such properties.

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Table 7. Parameters of some compositions of the (Na, Li, Cd0.5)NbO3 system No Composition 1 2 3 4 5

TC, K 638 668 673 693 703

ε33T/ε0 195 1070 295 520 415

Kp 0.13 0.28 0.14 0.22 0.18

g31, mV·m/N 5.1 6.2 4.4 5.4 5.1

Qm 670 225 560 340 560

VR, km/s 5.8 5.8 5.6 5.2 5.3

6.2. Solid Solutions of the (Na, Li, Sr0.5)NbO3 System Figure 16 shows a portion of the phase diagram adjacent to NaNbO3. Thin lines show the y- and z-sections and the thicker lines delineate the boundary lines of regions of different symmetry (one- and two-phase regions). It should be noted that the phase diagram of the system under study is of less complicated character than that described above. In the (Na, Li, Sr0.5)NbO3 system one can distinguish a wide MR1 (M2+Rh) and a narrow MR2 (Rh+M2) in the vicinity of which one should expect the appearance of extreme values of electrophysical and structural parameters.

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Figure 16. Phase diagram of the ternary system (Na, Li, Sr0.5)NbO3.

Let us consider the z-section which is the most typical section of the system under consideration. Figure 17 shows the concentration dependences of parameters for z = 15 mol. %; dashed lines delineate a wide MR1 and a narrow MR2. Figure 17 presents δ, ε33T/ε0, ε/ε0, Kp, d31, g31, tgδ, Qm, VR, YE11. It is seen that the dielectric parameters and the parameter g31 pass through the maxima at the opposite boundaries of MR1 which is typical of the ferroelectric systems. The positions of the Kp, d31 and ε33T/ε0 maxima coincide. Corresponding to them are the minimum δ values inside MR1. As to other parameters, the behavior of some of them correlates with the change in the degree of ferrostiffness of SS, namely: the parameters VR and YE11 vary in the direction opposite to the change of ε33T/ε0. However, the changes of tanδ and Qm cannot be explained from this point of view. It should be noted that their dependences exhibit kinks at the boundary between MR1 and the Rh phase.

Figure 17 (a).

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Figure 17 (b).

Figure 17 (c). Figure 17. Dependences of dielectric (a), piezoelectric (b), structural and elastic (c) characteristics of SS of the (Na, Li, Sr0.5)NbO3 system on the LiNbO3 content for the section z = 15 mol.% Sr0.5NbO3 (1 ε/ε0; 2 - ε33T/ε0; 3 - tanδ; 4 - Kp; 5 - d31; 6 - g31; 7 - Qm; 8 - VR; 9 - δM2; 10 - δRh; 11 - Y11E).

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Unlike the z-section, in other sections of this type one can observe considerably broken characteristics and this may be related not only to the MR position. Approximately in the same range of the LiNbO3 concentrations, where one can observe the maxima of ε33T/ε0, Kp, d31, g31, subsystems with a different degree of compositional ordering of various ions in Asublattices were revealed. To elucidate the effect of the degree of ordering of these ions on electrophysical parameters of SS under study we have analyzed the data of Reference [81] on the origin of high values of the dielectric permittivity in complex oxides A(B'1/2B'1/2)O3 with a disordered structure. In Reference [81] it was concluded that in an ordered structure die small B ions being in a regular surrounding by the larger ions have a considerably smaller free space for displacements than in a disordered structure. Therefore, on applying an electric field, the small ions shift much easier in the disordered structure without destruction of the oxygen framework than in the ordered one. This leads to a larger polarization per a unit of the electric field and, hence, to larger dielectric permittivity values. The same mechanism of the increase of the dielectric permittivity and the dependent electrophysical parameters may, in our judgement, account for the observed extremums on the boundaries of the above-mentioned systems with a different degree of ordering. One of the most interesting properties of individual SS of the system under consideration is the combination of rather low values of the dielectric permittivity (ε33T/ε0 = 105 ÷ 125) with a quite high electromechanical coupling factor (Kp = 0.2 ÷ 0.3), which leads to high values of the coefficient g31 characterizing sensibility to mechanical stresses. As known [14, 62], such materials may be effectively used in accelerometers, defectoscopes and diagnostic medical equipment. The low dielectric permittivity is favorable for these materials to be utilized in high-frequency transducers. Table 8 summarizes a number of compositions with such properties.

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Table 8. Parameters of some compositions of the (Na, Li, Sr0.5)NbO3 system No Composition 1 2 3

TC, K 593 611 560

ε33T/ε0 127 107 110

Kp 0.193 0.215 0.296

g31, mV·m/N 10.0 13.7 15.8

Qm 650 40 295

VR, km/s 5.7 4.9 5.4

The precision X-ray studies of SS of the ternary systems (Na, Li, Cd0.5)NbO3 and (Na, Li, Sr0.5)NbO3 enabled us to determine more accurately the symmetry of crystallizing phases, structural transitions and the morphology of morphotropic regions. The dependences of electrophysical parameters of SS of the above-mentioned systems were studied in a wide range of concentrations, and their relation with structural parameters, in particular, with the homogeneous deformation parameter was established. Compositions were obtained with unique combinations of electrophysical parameters (low values of density, high values of sound speed and piezoelectric parameters for a broad spectrum of dielectric permittivity values) useful for various applications (HF-transducers, defectoscopes, accelerometers, diagnostic medical equipment). The Table 9 lists the main characteristics of some Pb-free ferroelectric ceramic materials designed at the Research Institute of SFU.

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Table 9. The main characteristics of some Pb-free FPCM

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Materials

Main parameters ТC, K ε33Т/ ε0

Кр

Fields of application

PCR – 34 based on (Na, K)NbO3 PCR – 35 based on (Na, Li)NbO3 PCR – 90 based on (Na, Li, Sr0.5) NbO3

690

460

640

‫׀‬g31,‫׀‬ mV∙m/N 10.0

Qm

0.42

‫׀‬d31,‫׀‬ pC/N 45

150

VR, km/s 5.40

120

0.22

12

10.5

1000

5.90

610

100

0.30

15

16.9

245

5.95

PCR – 61 based on LiNbO3

> 1470

48

0

0

0

400

5.2

PCR – 92 based on (Na, Li, Cd0.5) NbO3

590

2700

0.2

189

7.3

360

5.50

High-frequency acoustoelectric transducers Devices on surface acoustic waves High-sensitivity accelerometers, flaw detectors, medical diagnostic equipment High-temperature pressure indicators Low- and centerfrequency transducers

It should be noted that, despite a great number of works on the Pb-free ceramics published for the last three years, none of the countries taken the path of engineering the ecologically pure products (materials) and technologies succeeded in solving this problem [82-91] because to produce the Pb-free FPCM having a necessary set of parameters is a success only on using the exotic, labour- and power- consuming methods. Unfortunately, it is impossible to use such methods on a commercial scale. We undertook the task of working out a mass (serial) technology of manufacturing Pbfree materials based on the conventional ceramic technology adapted to the specific SS. The results of investigations of the multicomponent composition based on SS of the (Na,K)NbO3 system with additives are presented below.

7. PRODUCTION AND DIELECTRIC PROPERTIES OF LEAD-FREE CERAMICS WITH THE FORMULA [(Na0.5K0.5)1 – XLiX](Nb1 – Y –ZTaYSbZ)O3 The basis for most ferro-piezoceramic materials used in production at present are solid solutions (SS) of lead containing systems PbTiO3 - PbZrO3, Pb(Nb2/3Mg1/3)O3 - PbTiO3, etc. The most widespread technology of their production includes solid-phase synthesis and agglomeration at high temperatures. In light of the significant toxicity of lead compounds in recent years, the search for alternative materials is being carried out [83]. Especially important is elimination of lead compounds from the composition of special electrotechnical ceramics (piezoceramics) [1]. Thus, study of the possible use of ecologically clean lead-free materials in the field of electrical engineering is of great importance.

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Attempts to obtain the necessary properties by means of developing multicomponent systems are described in [12, 92, 93]. Works [94, 95] consider use of rather exotic methods aiming to obtain alkali metal (AM) niobates: topochemical texturing and synthesis in plasma discharge. The results of the works performed are quite contradictory and, moreover, the best of them have been obtained with the use of labor-intensive technologies that can be ascribed only to laboratory ones. The objective of this work is the development of new lead-free materials and a method of their production on the basis of conventional ceramic technology.

7.1. Objects of Investigation Multicomponent SS [(Na0.5K0.5)1–xLix] (Nb1–y–zTaySbz)O3 at x = 0 ÷ 0.14, y = 0, 0.1, 0.2, and z = 0 ’ 0.1 with the step Δz = 0.01 were studied. The work included three steps: 1) Synthesis, agglomeration, and investigation of SS of eight sections. (1 – x)(Na0.5K0.5)NbO3–xLiSbO3 (I), (1 –x) (Na0.5K0.5)NbO3–xLiTaO3 (II), [(Na0.5K0.5)1 –xLix](Nb0.9Ta0.1)O3 (III), (1 –x) (Na0.5K0.5)(Nb0.94Ta0.06)O3–xLiTaO3 (IV),

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[(Na0.5K0.5)1 –xLix][(Nb0.9Ta0.1)1 –xSbx]O3 (V), (1 –x)(Na0.54K0.46)(Nb0.9Ta0.1)–xLiSbO3 (VI), [(Na0.5K0.5)1 –xLix](Nb0.8Ta0.2)O3 (VII), (1 –x)(Na0.5K0.5)(Nb0.84Ta0.16)O3–xLiTaO3 (VIII). Figure 18 shows the concentration field of the studied compounds. 2) Search for and use of more stable raw materials in order to obtain new SS. Initial carbonates were changed to hydrocarbonates in order to decrease hydrolysis of the reaction mixture during synthesis and retain stoichiometry of the preset compounds of materials. 3) Selection and introduction of various modifiers with the goal of improving SS electrophysical properties. For this purpose, we used compounds of two sections (VI and VIII), since according to the preliminary research [96], the respective SS had the best set of dielectric and piezoelectric parameters. SS of section VI in addition to stoichiometry were modified by oxides of strontium and titanium (SrO + TiO2) in the amount of 1.4 mol %, which resulted in obtaining SS with the formula

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[(1 – x)(Na0.54K0.46)(Nb0.9Ta0.1)O3–xLiSbO3] + 0.014(SrO +TiO2). SS of section VIII were modified in three ways. In the first case, cations of alkali metals were replaced by calcium, which was added as CaCO3 in the amount of 2 at. %, and the formula took the form (1 – x)(Na0.48K0.48Ca0.02)(Nb0.84Ta0.16)O3–xLiTaO3 In the second case, apart from replacement by Ca in sublattice A, we injected LiSbO3 as a second component of the quasibinary section of the multicomponent system instead of LiTaO3. The formula of SS took the form (1 – x)(Na0.48K0.48Ca0.02)(Nb0.84Ta0.16)O3–xLiSbO3 In the third case, we used superstoichiometric additions: 1 mol. % CdO and 2 mol. % (CuO + TiO2). The structure of SS in this case can be described by the following formulas: [(1 – x)(Na0.5K0.5)(Nb0.84Ta0.16)O3 –xLiTaO3] + 0.01CdO, [(1 – x)(Na0.5K0.5)(Nb0.84Ta0.16)O3 –xLiTaO3] + 0.02% (CuO + TiO2).

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7.2. Methods of Sample Production The main difficulty in production of such materials is to achieve high ceramic density. Hydrolysis of alkali metal compositions during SS synthesis results in formation of easily melted, volatile, very aggressive media, in particular, hydroxides of AM with a melting temperature much lower than ceramic agglomeration temperatures (Tmelt (KOH) = 670 K, Tmelt (NaOH) = 590 K) and forming a liquid phase at agglomeration.

Figure 18. Concentration field of compositions [(Na0.5K0.5)1–xLix](Nb1–y–zTaySbz)O3 and sections I-VIII.

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At the first stage, SS were synthesized by the method of solid phase reaction in two ways: from carbonates of AM (sections III, V, VII) and from preliminarily synthesized SS bases (Na1–xKx)(Nb1–yTay)O3, LiTaO3, LiSbO3, and LiSbO3 (sections I, II, IV, VI, VIII). To achieve the maximum attainable reaction, the synthesis of SS bases, SS sections III, V, VII, and LiTaO3 and LiSbO3 was carried out in two stages with intermediate grinding in ethanol. The temperatures of synthesis were selected by a series of trial burnings with controlled reaction completeness and phase composition of the products by the RFM method. The temperatures and duration of burning were T1 = T2 = (1120 ÷ 1140) K (depending on the compound) and η1 = η2 = 6 h. Final SS of sections I, II, IV, VI, VIII were synthesized by one-stage burning from preliminarily synthesized SS bases and compositions of LiTaO3 and LiSbO3 at 1140 K for 6 h. According to RFM, not in any case of admixtures were hydrolysis products found in the synthesized powders. The agglomeration temperature was (1380 ÷ 1430) K (depending on the composition), and the time of agglomeration was (1 ÷ 2) h (depending on the compound).

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7.3. Methods of Sample Analysis The sample density was determined by the method of hydrostatic weighing in octane. Reaction completeness and phase composition of the product were controlled by RFM method (DRON-3, FeKα radiation). Dielectric spectra were studied on a YuCOMP 2.0 laboratory stand (produced at the Scientific Research Institute of Physics, Southern Federal University) with the use of an E7-20 immitance measuring device. Measurements were taken in the range of temperatures (295 ÷ 970) K and frequencies 25 Hz ÷ 1 MHz. The dielectric, piezoelectric, and elastic characteristics were measured at room temperature in accordance with AUS 11 0444 87. At the same time, the relative dielectric permeability of polarized (ε33T/ε0) and nonpolarized (ε/ε0) samples, piezomodulus (|d31|), coefficient of electromechanical bond of planar oscillation mode (Kp), mechanical quality factor (Qm), Young‘s modulus (YE11) and sound velocity (V1E) were determined. Ceramic samples durable under mechanical treatment in water were obtained for compositions containing tantalum at lithium concentration x = 0.02 ÷ 0.08. The relative density of the samples is ρrel = ρexp/ρren ≥ 90%. Antimony present in these compositions also improves ceramic agglomeration. Figure 19 demonstrates the most specific dielectric spectra of samples obtained at the first stage of the work. The samples in general are characterized by high conductivity (δ) and considerable, especially at low frequencies, dispersion of ε/ε0 (Figure 19). This fundamentally restricts the area of application of the given ceramic compositions. Nevertheless, a number of compounds belonging to sections II, VI, and VIII have interesting properties. So, in the case of the compositions of section II, there are relatively high values ε/ε0: both upon heating and upon cooling, a sharp maximum is observed, which proves the stability of the respective SS (Figures 19a, 19b, curves 1). The largest maximum values ε/ε0 are specific to compositions belonging to sections II and VI. In the samples of section VI, upon heating, there is a very high and unclear extended maximum ε/ε0, caused by high δ. Upon sample heating to 970 K, defects were annealed and moisture partially removed, which caused δ to decrease drastically and the dielectric spectrum to take a form that is more specific to ferroelectric materials with a sharp maximum ε/ε0 and its weak dispersion (Figure 19a, curves 3). It is worth noting that, upon cooling, samples of section VI had higher values of maximum ε/ε0 and less dispersion than those of section II. Comparable

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although smaller values were obtained for the compositions of section VIII. In our opinion, significant influence on the properties of the ceramic is exerted by the processes of its degradation due to hydrolysis and, as a result, high δ. Comparison of the character of dielectric spectra for SS with different compositions demonstrates that annealing of defects and degradation of the conductive layer result in a substantial decrease in δ. This becomes strikingly apparent in the compositions containing antimony (section VI). Figure 20 shows dielectric spectra of SS obtained at the second stage of the work. The replacement of alkali metal carbonates by hydrocarbonates serves to substantially decrease the level of δ. This is readily apparent in Figure 20a: in temperature dependences ε/ε0 obtained even at low frequencies, there is a sharp maximum, which for the samples produced from carbonates is observed in the background of δ only at very high frequencies (Figure 20a, inset).

Figure 19. Dielectric spectra of samples of sections II, VI, VIII: (a) upon heating (direct reaction), (b) upon cooling (reverse reaction) (1) (1 – x)(Na0.5K0.5)NbO3–xLiTaO3 (x = 0.12), section II; (2) (1– x)(Na0.5K0.5)(Nb0.84Ta0.16)O3–xLiTaO3 (x = 0.05), section VIII; (3) (1–x) (Na0.54K0.46)(Nb0.9Ta0.1)– xLiSbO3 (x = 0.04), section VI).

The effect of modifiers is ambiguous. So, injection of CaO, CdO, and CuO + TiO2 into the compositions of section VIII results in decreased δ and dispersion due to it ε/ε0 (Figures

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20b, 20e, 20f). In addition, in the case of CaO modification, there is a significant increase in the maximum ε/ε0 measured at high frequencies, which is undoubtedly tied to the decreased δ. Upon replacement of LiTaO3 by LiSbO3, injection of Ca results in increased δ (Figure 20d). Concerning additions of Sré to the compositions of section VI, their injection serves to increase δ and displace the peak ε/ε0 (especially at high frequencies) into an area with lower temperature; at the same time, its extension and decrease of the respective peak value ε/ε0 occur (Figure 20b). As is apparent from Table 10, the best piezoproperties are specific to samples of sections II, VI, and VIII. At the same time, for sections VI and VIII, a rather high repeatability of results has been achieved.

Figure 20. Dielectric spectra of ceramic spectra: (a) section VI (x = 0.02), obtained at the use of new raw materials; (b) section VI (x = 0.02), modified by SrO + TiO2; (c, d) section VIII, modified by CaO (x = 0.06) ((d) second method of modification); (e, f) section VIII (x = 0.06), modified by CdO and CuO + TiO2, respectively of δ only at very high frequencies (Figure 20a, inset).

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Table 10. Electrophysical parameters of SS [(Na0.5K0.5)1–xLix](Nb1–y–zTaySbz)O3 measured at room temperature (first stage of work) Compositions

ε33T/ε0

tgδp.p.

Kp

‫׀‬d31,‫ ׀‬pC/N

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(1-x)(Na0.5K0.5)NbO3 – xLiTaO3 (section II) x = 0.05 447 0.297 0.22 31.6 x = 0.07 416 0.451 0.23 31.2 x = 0.08 476 0.636 0.19 27.9 x = 0.12 459 0.155 0.18 25.6 x = 0.14 613 0.102 0.2 27.8 [(Na0.5K0.5)1-xLix](Nb0.9Ta0.1)O3 (section III) x = 0.06 898 0.501 0.1 20.5 (1-x)(Na0.5K0.5)(Nb0.94Ta0.06)O3 – xLiTaO3 (section IV) x = 0.08 2518 0.389 0.06 16.8 [(Na0.5K0.5)1-xLix][(Nb0.9Ta0.1)1-xSbx]O3 (section V) x = 0.06 783 0.205 0.1 17.7 x = 0.04 879 0.366 0.09 16.1 (1-x)(Na0.54K0.46)(Nb0.9Ta0.1) – xLiSbO3 (section VI) x = 0.02 543 0.055 0.21 29.7 x = 0.03 733 0.1 0.2 31.9 x = 0.1 494 0.105 0.26 34.5 [Na0.5K0.5)1-xLix](Nb0.8Ta0.2)O3 (section VII) x = 0.1 954 0.318 0.14 25.6 (1-x)(Na0.5K0.5)(Nb0.84Ta0.16)O3 – xLiTaO3 (section VIII) x = 0.05 658 0.222 0.18 26.2 x = 0.08 749 0.128 0.17 25.9

Qm

VЕ1, km/s

YE11 ∙ 10-11 N/m2

63 67 56 101 95

3.86 3.82 3.81 3.93 4.43

0.599 0.585 0.603 0.616 0.825

27

4.04

0.677

15

4.31

0.821

48 54

4.33 4.34

0.758 0.842

92 62 96

4.27 4.31 4.26

0.835 0.853 0.796

18

4.12

0.774

43 75

4.16 4.1

0.818 0.809

In the course of the second and third stages of the work as a result of using the new raw material (hydrocarbonates of alkali metals) and modifiers, we have achieved some improvement of electrophysical properties (Table 11). As is apparent from Table 11, at the second and third stages, we have managed to achieve substantial (on average an order of magnitude) decrease in δ, which is obviously tied to the decreased degree of hydrolysis during synthesis. In a number of cases, in samples of SS, we have achieved noticeably higher piezoelectric parameters and observed some growth of Qm, which indicates the higher quality of samples. A method for production of lead-free ferro-piezoceramic materials has been developed. It consists in selection of chemical compositions, selection of stable raw materials (hydrocarbonates of alkali metals), synthesis of final products from preliminarily obtained SS bases, agglomeration of small intermediate parts (with diameter of 12 mm and thickness of 3 mm) at one layer feeding, use of corundum substrates and crush-rock pads containing zirconates of alkali metals and separating the reaction mixture from the substrate, clear regulation of synthesis and agglomeration modes, selection and introduction of modifiers, and application of optimal conditions of polarization.

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Table 11. Electrophysical parameters of SS [(Na0.5K0.5)1–xLix](Nb1–y-zTaySbz)O3 obtained with the use of hydrocarbonates of alkali metals and a number of modifiers measured at room temperature (second and third stages of work) Compositions

ε33T/ε0

tgδ

Kp

‫׀‬d31N/Cp ,‫׀‬

Qm

VЕ1, km/s

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(1-x)(Na0.48K0.48Са0.02)(Nb0.84Ta0.16)O3-xLiTaO3 (section VIII) x = 0.06 701 0.0377 0.09 17.8 64 4.5 x = 0.08 661 0.0312 0.13 24.4 28 4.6 x = 0.1 545 0.0352 0.10 18.7 22 4.63 [(1-x)(Na0.5K0.5)(Nb0.84Ta0.16)O3 - xLiTaO3] + 0.01СdO (section VIII) x = 0.06 714 0.0267 0.29 62.3 91 4.34 x = 0.08 551 0.0311 0.18 32.2 63 4.53 [(1-x)(Na0.5K0.5)(Nb0.84Ta0.16)O3 - xLiTaO3] + 0.02 % (СuO + TiO2) (section VIII) x = 0.02 259 0.0089 0.26 38.3 68 4.36 x = 0.04 282 0.0147 0.22 33.3 145 4.22 x = 0.06 346 0.0527 0.21 30.8 243 4.69 x = 0.08 267 0.0138 0.13 20.5 75 3.52 x = 0.1 257 0.0195 0.12 20.3 93 3.95 (1-x)(Na0.54K0.46)(Nb0.9Ta0.1) – xLiSbO3 (from other initial substances) (section VI) x = 0.02 368 0.0248 0.32 51.1 133 4.54 x = 0.04 330 0.0271 0.34 54.0 51 4.45 x = 0.06 452 0.0275 0.28 58.3 102 3.93 x = 0.08 486 0.0318 0.19 43.8 70 3.67 x = 0.1 434 0.0321 0.17 39.0 74 3.61 [(1-х)(Na0.54K0.46)1-x[(Nb0.9Ta0.1)O3 – LiSbO3] + 0.014( SrO + TiO2) (section VI) x = 0.04 495 0.0804 0.03 67.2 48 x = 0.06 523 0.0326 0.06 13.9 64 -

YE11 ∙ 10-11 N/m2 1.017 1.02 1.086 0.928 0.999 0.814 0.783 0.973 0.555 0.661 0.897 0.842 0.656 0.574 0.522 -

These technological methods have served to achieve values Kp ≥ 0.34 and d31 ≥ 70 pC/N at a low value of ε33T/ε0 ~ 300 and rather high velocity ~ 4.5 km/s. Among modified SS, the best properties are specific to the samples modified by CdO and CuO + TiO2 of section VIII. At the same time, the most prospective are SS of section VI with 0.02 ≤ x ≤ 0.06, for which the best electrophysical parameters have been obtained: Kp ≥ 0.34, d31 ≥ 50 pC/N, and VE1 ~ 4.5 km/s at ε33T/ε0 ~ 330.

8. DEFORMATION, POLARIZATION, AND REVERSIBLE PROPERTIES OF LEAD-FREE CERAMICS BASED ON ALKALI METAL NIOBATES The fabrication of ferroelectric/piezoelectric (FP) ceramic materials is a high-technology process. At the same time, it may present serious environmental problems. In today‘s world, environmental considerations are of key importance in deciding whether a given material can be used on a commercial scale. One major cause of environment pollution is wide use of lead compounds, which are present as a key component in most commercial FP ceramics.

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Therefore, one of the challenges for modern materials research is to find a safe alternative to lead-containing FP ceramics. In recent years, alternative materials have been the subject of intense attention. Previous work [82, 85, 97–98] has shown that the binary system NaNbO3 – KNbO3 is the best starting point for designing ceramics capable of replacing lead-containing FP ceramics. Solid solutions of this system are similar in piezoelectric properties to some PZT ceramics. At the same time, the polarization and reversible properties of FP ceramics based on alkali metal niobates have not yet been studied. The purpose of this work was to investigate the properties of [(Na0.5K0.5)1–xLix](Nb1–y–zTaySbz)O3 in a composition region near the binary system NaNbO3 – KNbO3. The sample preparation procedure was similar to that described previously [98]. Preliminary studies of the electrical properties of samples along eight cuts (1 – x)(Na0.5K0.5)NbO3–xLiSbO3 (I), (1 –x) (Na0.5K0.5)NbO3–xLiTaO3 (II), [(Na0.5K0.5)1 –xLix](Nb0.9Ta0.1)O3 (III), (1 –x) (Na0.5K0.5)(Nb0.94Ta0.06)O3–xLiTaO3 (IV), [(Na0.5K0.5)1 –xLix][(Nb0.9Ta0.1)1 –xSbx]O3 (V), (1 –x) (Na0.54K0.46)(Nb0.9Ta0.1)–xLiSbO3 (VI), [Na0.5K0.5)1 –xLix](Nb0.8Ta0.2)O3 (VII),

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(1 –x) (Na0.5K0.5)(Nb0.84Ta0.16)O3 – xLiTaO3 (VIII), allowed us to identify the most promising materials [98].The best properties were offered by solid solution VI and VIII. They were modified to give the following materials for this investigation: (1 –x) (Na0.54K0.46)(Nb0.9Ta0.1) – xLiSbO3, [(1 –x) (Na0.54K0.46)1 –x[(Nb0.9Ta0.1)O3–LiSbO3] + 0.014(SrO+TiO2), (1 –x) (Na0.48K0.48Ca0.02)(Nb0.84Ta0.16)O3 – xLiTaO3, (1 – x) (Na0.48K0.48Ca0.02)(Nb0.84Ta0.16)O3 – xLiSbO3, [(1 – x)Na0.5K0.5)(Nb0.84Ta0.16)O3–xLiTaO3] + 0.01CdO, [(1 – x) (Na0.5K0.5)(Nb0.84Ta0.16)O3 –xLiTaO3] + 0.02% (CuO + TiO2). Solid solutions were synthesized by solid-state reactions using two procedures: from alkali metal carbonates (cuts III, V, and VII); from presynthesized (Na1 – xKx)(Nb1 – yTay)O3, LiTaO3 solid solutions, LiTaO3, and LiSbO3 (cuts I, II, IV, VI, and VIII). To drive reactions

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to completion, synthesis was carried out in two steps at (1120 ÷ 1140) K for η1 = η2 = 6 h with one intermediate grinding. The resultant materials were sintered at (1380 ÷ 1430) K for (1 ÷ 2) h, depending on composition. The phase compositions of the samples and the completion of reactions were checked by powder X-ray diffraction (XRD) on a DRON-3 diffractometer (filtered CoKα radiation, Bragg–Brentano geometry). The strain ξ3 induced by an applied electric field E3 was measured with a purpose-designed system, which enabled quasi-static measurements of the sample size. The ξ3(E3) half-loops obtained were used to plot the piezoelectric voltage coefficient d33 versus applied electric field E3 [99]. Dielectric hysteresis loops were measured by an oscilloscope method (f = 50 Hz). We determined the total, induced, reorientational, and residual polarization [124]. The reversible relative dielectric permittivity of unpoled (ε/ε0) and poled (ε33T/ε0) samples was measured using a system designed at the Research Institute of Physics, Southern Federal University. Its operating principle was described elsewhere [100]. Figures 21 and 22 show the dc electric field dependences of the piezoelectric voltage coefficient d33 and electromechanical hysteresis half-loops ξ3 typical of samples with high electrical strength (cut VI, x = 0.02, 0.06, 0.1; cut VIII with CdO, CuO + TiO2, and CaO additions). In the electric field range studied (0 < E < (11 ÷ 15) kV/cm), all of the solid solutions have monotonic ξ3(E) curves under both forward and reverse bias. The d33(E) curves for the solid solutions in different cuts are similar in shape to those for medium-hard ferroelectric materials [12]. In particular, the d33 of solid solutions VI rises slightly (by 15 ÷ 30%) for E < 9.0 kV/cm and either decreases slightly (by 5 ÷ 9%) or remains constant at E above ~ 9 kV/cm. Increasing the LiSbO3 content (x) impairs the piezoelectric performance of the samples (Figure 21d), and the d33(E) curve approaches that typical of hard ferroelectric materials (linear, almost field-independent). The d33 of solid solutions VIII (Figure 22) rises linearly in weak fields (increases by 15 ÷ 30%) and has a plateau at Е = (7.0 ÷ 9.0) kV/cm, where the piezoelectric modulus either decreases very slightly or remains constant. Note that solid solutions VIII modified with CuO + TiO2 (Figure 22b) are the hardest ferroelectrics. Solid solutions VIII modified with CdO (Figure 22a) and CaO (Figure 22c) are softer ferroelectrics. Solid solutions VI are still softer. Figures 23 and 24 show the most typical dielectric hysteresis loops. Their nonlinear, nearly rectangular shape, with a significant remanent polarization suggests that the materials are ferroelectrics. In particular, the polarization versus electric field curves of solid solutions VI with low LiSbO3 contents (x) are the most similar in shape to classic loops (Figure 23a). With increasing x, the loops become slimmer, which is accompanied by a sharp drop in polarization, an increase in its induced component, and an increase in coercive field (Figures 23b–23d). These effects are probably associated with a reduction in the percentage of low-symmetry phases in the solid solutions [96]. Figure 24 shows the dielectric hysteresis loops of samples VIII modified with CuO + TiO2 (x = 0.06) (Figure 24a), CdO (x = 0.04) (Figure 24b), and CaO (x = 0.06) (Figure 24c). The P(E) curves in Figures 24a and 4c were obtained for samples with the same LiTaO3 content (x) but different additions, which allows us to compare the effect of those additions on the shape of the dielectric hysteresis loop. The loop in Figure 24a (CuO + TiO2) is flatter and has a markedly higher coercive field than does the loop in Figure 24c (CaO) even though they have almost the same remanent polarization. It seems likely that the shape of the loop in Figure 24a can be accounted for by an increase in ferroelectric hardness and increased electrical conductivity.

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Figure 21. (1) Piezoelectric voltage coefficient d33 as a function of dc electric field and (2) electromechanical hysteresis half-loops ξ3 for ceramics of cut VI at LiSbO3 contents x = (a) 0.02, (b) 0.06, and (c) 0.1. (d) Maximum piezoelectric modulus as a function of (1) x and (2) the corresponding electric field.

Figure 22. (1) Piezoelectric voltage coefficient d33 as a function of dc electric field and (2) electromechanical hysteresis half-loops ξ3 for ceramics of cut VIII modified with (a) CdO, (b) CuO + TiO2, and (c) CaO.

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Figure 23. Dielectric hysteresis loops of samples VI with x = (a) 0.04, (b) 0.06, (c) 0.08, and (d) 0.1.

Figure 24. Dielectric hysteresis loops of samples VIII modified with (a) CuO + TiO2 (x = 0.06), (b) CdO (x = 0.04), and (c) CaO (x = 0.06).

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The curve P(E) in Figure 24b is similar in shape to those of classic FP ceramics and has a considerably higher remanent polarization in comparison to those described above. Ceramics VIII modified with CuO + TiO2 have the highest coercive field Ec and rather low polarization, which indicates that they are the hardest ferroelectrics among the materials studied. Ceramics VI have rather low Ec (Figure 23), medium dielectric permittivity, and the highest piezoelectric performance. Ceramics VI have extrema in their properties at x = 0.04: a sharp peak in polarization and a slight reduction in Ec. Such behavior may be due to a morphotropic phase transition, as confirmed by XRD results [96]. Figures 25–27 plot reversible ε/ε0 and ε33T/ε0 versus dc electric field. These data were analyzed based on previous results [12, 99, 101–106]. As seen in Figures 25–27, all of the curves, except those of samples VIII, which will be discussed below, are butterfly- shaped, which is typical of classic FP ceramic materials [104]. The relatively low ε/ε0 values, weakly dependent on electric field, suggest that these materials are hard FP ceramics [106]. The same is evidenced by the small slope of the ε/ε0(E) and ε33T/ε0(E) curves under both forward and reverse bias, which leads to broadening of the maximum in permittivity. At the same time, the low fields corresponding to the maximum in samples VI are reminiscent of soft FP ceramics [106]. This is probably due to opposite effects of a few factors. On the one hand, 180° domain switching occurs more readily in the initial stage, leading to a reduction in ε/ε0 and ε33T/ε0 because of the weaker effect of hindrances in the presence of extended domain walls. On the other hand, the domain wall mobility in niobate ceramics may be reduced because of the considerable defect density. This, in turn, may significantly hinder 180° domain reorientations, thereby reducing ε/ε0 in high fields. Note that ceramics VIII modified with CuO + TiO2 are the hardest ferroelectrics among the materials studied (Figure 26). They have low ε/ε0 and ε33T/ε0 values, which vary only slightly with electric field. The reversible permittivity versus field curves of poled samples are asymmetric: the reverse-bias curves exhibit significant hysteresis, have a broad maximum in ε33T/ε0, and lie above the forward-bias curves, which are almost linear and have lower ε33T/ε0 values. This behavior of reversible ε33T/ε0 points to significant anisotropy in domain wall mobility along the polar axis, which may be related to the directional growth of domains in an applied field during poling and a sharp rise in the stability of domain walls due to the increased defect density in the materials. The formation of large, stable domains blocks the motion of those domains with the opposite polarization direction. As a result, the domain structure is considerably more mobile in the volume polarization direction than in the opposite direction, and its mobility seems to be determined by the growth of large, bulky domains with pinned walls as a result of poling. With increasing x (LiTaO3 content), ceramics VIII modified with CuO + TiO2 show weaker hysteresis in ε/ε0 and stronger anisotropy resulting from poling (Figure 26b). These effects indicate that the materials become harder ferroelectrics. The highest ε/ε0 is offered by solid solutions VIII modified with CaO (Figure 27), but their ε/ε0 is a weak function of applied electric field and is determined in large part by the conductivity of the ceramics [98]. Poling of these ceramics produces no drastic changes in the behavior of their reversible permittivity: it retains a symmetric butterfly shape and remains a weak function of applied electric field. Samples VI are the softest ferroelectrics among the solid solutions studied (Figure 25): they have medium ε/ε0 values and are considerably more sensitive to an applied electric field. In this case, poling leads to the formation of sharper maxima in ε33T/ε0, broadens the hysteresis in ε33T/ε0, and makes the loops more asymmetric.

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Figure 25. Reversible dielectric permittivity as a function of dc electric field for (a) unpoled (ε/ε0) and (b) poled (ε33T/ε0) samples VI.

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Figure 26. Reversible dielectric permittivity as a function of dc electric field for (a) unpoled (ε/ε0) and (b) poled (ε33T/ε0) samples VIII modified with CuO + TiO2.

Figure 27. Reversible dielectric permittivity as a function of dc electric field for (a) unpoled (ε/ε0) and (b) poled (ε33T/ε0) samples VIII modified with CaO.

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Consider in greater detail the composition effect on the reversible characteristics of ceramics VI (Figure 25). The solid solutions can be divided into two groups, differing in both structural and reversible characteristics. One group comprises the materials with x < 0.04, which consist of a mixture of low-symmetry phases and are harder ferroelectrics. The other group is constituted by the ceramics with x > 0.04, which contain tetragonally distorted phases. They have high ε/ε0 values and increased sensitivity to an applied electric field. The solid solution with x = 0.04 warrants special attention, because its composition lies in the morphotropic region: before and after poling, its behavior is similar to that of the materials of the second and first groups, respectively. This composition has extrema in ε/ε0, with increased sensitivity to an applied electric field. The environmentally friendly, lead-free materials studied are hard FP ceramic materials with low dielectric permittivity. The introduction of CuO + TiO2 and CdO raises the ferroelectric hardness of the lead-free FP ceramics and improves their polarization and deformation characteristics, without adversely affecting their dielectric properties. Solid solutions VI with 0.02 ≤ x ≤ 0.04 are the most attractive for producing specialty electroceramics: P = 4.698 μC/cm2, Ec = 4.17 kV/cm (f = 50 Hz), and d33max ~ 500 pC/N.

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9. CONCLUSIONS We succeeded in obtaining the Pb-free materials the advantages of which are as follows: - the high sound velocity (for a thickness mode of vibration the frequency constant Nt = 3000 kHz∙mm, i.e. V3D = 6000 m/s) defining the high-frequency range of operation of the transducer and, also, enabling one to obtain the prescribed frequency on less thin plates that simplifies the technology of making HF-devices at the expense of a possibility of increasing their resonance sizes (this, in turn, is favourable from the standpoint of decreasing the transducer capacity); - the low density (4.5 g/cm3) resulting, from the one hand, in the considerably decreased weight of the products and, from the other hand, in the decreased acoustic impedance (Za = ρV3D = 27∙106 kg/m2∙s (27 mrayl) which is necessary for matching with the acoustic load; - the very low dielectric permittivity (ε33T/ε0 < 100) that is of small importance for the electric matching with a generator and a load; - the increased thickness electromechanical coupling factor (Kt = 0.48 ÷ 0.51) characterizing the effectiveness of operation of a transducer in the echo-regime and in the receiving regime; - in some cases, the infinite anisotropy of piezoelectric properties (Kt/Kp → ∞, d33/‫׀‬d31 ‫׀‬ → ∞) enabling one to improve a signal-to-noise ratio and to simplify the technology by eliminating the operation of cutting the material into subelements; - the low dielectric – tgδ and moderate mechanical – 1/Qm losses which are important for obtaining the short pulses and the uniform amplitude-frequency characteristics. It should be noted that a reproducibility of properties of the niobate ceramics is not inferior to that of PZT-materials and is considerably superior to that of parameters of the materials based on Pb titanate and metaniobate.

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In addition to the compositions discussed above, we have discovered the promising chemical compositions of interest in connection with the search for relaxor Pb-free materials. The possibility was ascertained of fabricating ferroelectric lithium niobate films (by HFcathode sputtering) which distinguish themselves by the spontaneous polarization during their growth. Porous and composite AMN-based materials for application in hydroacoustics were engineered.

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[73] Reznichenko, L. A.; Dantsiger, A. Ya.; Razumovskaya, O. N.; et al. Tech. Phys. 2000, vol 45, 1437. [74] Freidenfel‘d, E. Zh.; Dambekalne, M. Ya.; Yanson, G. D. In Proc. III Interinstitution Conf. on Methods of Production and Analysis of Ferrite, Ferroelectric and Piezoelectric Materials and Related Raw Materials; Donetsk, 1970, p. 3. [75] Shilkina, L. A.; Pozdnyakova, I. V.; Reznichenko, L. A.; et al. In Proc. 8th Int. Symp. Physics of Ferroelectric-Semiconductors, Rostov-on-Don, 1998, p. 190. [76] Fesenko, E. G.; Dantsiger, A. Ya.; Reznichenko, L. A.; et al. Sov. Phys. Tech. Phys. 1982, vol 27, 1389. [77] Ivanova, L. S.; Reznitchenko L. A.; Razumovskaya O. N.; et al., Bul. of Acad. Sci SSSR, Inorg. Mater. 1987, vol 23, 525. [78] Lewis, B.; White, E. A. D., J. Electronics. 1956, vol 1, 646. [79] Tennery, V .J. J. Amer. Ceram. Soc.1966, vol 49, 376. [80] Fesenko, E. G.; Filipiev, V. S.; Kupriyanov, M. F. Phys. Stat. Sol., 1969, vol 11, 466. [81] Nomura, S.; Uchino, K. Ferroelectrics. 1982, vol 41, 117. [82] Saito, Y.; Takao, H.; Tani, T.; et al. Nature, 2004, vol 432, 84-87. [83] Cross, E. Nature, 2004, vol 432(4), 24-25. [84] Li, J.-F.; Wang, K.; Zhang, B.-P.; Zhang, L.-M. J. Am. Ceram. Soc. 2006, vol 89, 706709. [85] Tani, T. J. Korean Phys. Soc. 1998, vol 32, 1217-1220. [86] Shrout, Th. R.; Zhang, Sh. J. J. Electroceram. 2007, vol 19, 111-124. [87] Bomblai, P.; Sukprasert, S.; Muensit, S.; Milne, S. J. J. Mater. Sci. 2008, vol 43, 61166121. [88] Wang, C., Hou, Y.; Ge, H.; et al. J. Crystal Growth. 2008, vol 310, 4635-4639. [89] Tanaka, K.; Kakimoto, K.; Ohsato, H. J. Crystal Growth. 2006, vol 294, 209-213. [90] Du, H.; Li, Z.; Tang, F.; et al. Mater. Sci. Eng. 2006, vol 131, 83-87. [91] Kakimoto, K.; Masuda, I.; Ohsato, H. J. Europ. Ceram. Soc. 2005, vol 25, 2719-2722. [92] Ivliev, M. P.; Raevskii, I. P.; Reznitchenko, L. A.; et al. PTT, 2003, vol 45, 1886–1891. [93] Lin, D.; Kwok, K. W.; Lam, K. H.; et al. J. Phys. D: Appl. Phys. 2007, vol 40, 3500– 3505. [94] Yang, Z.; Chang, Y.; Wei, L. Appl. Phys. Lett. 2007, vol 90, 042911– 13. [95] Guo, Y.; Kakimoto, K.; Ohsato, H. Mater. Lett. 2005, vol 59, 241–244. [96] Verbenko, I. A.; Razumovskaya O. N.; Shilkina L. A. In Mater. X Int. Interdiscipl. Symp. “Order, disorder and oxide properties (ODPO–2007)”, Rostov-on-Don, 2007, pp 140–144. [97] Reznichenko, L. A.; Shilkina, L. A.; Razumovskaya, O. N.; et al. Inorg. Mater. 2003, vol 39, 139–150. [98] Verbenko, I. A.; Reznichenko, L. A.; Razumovskaya, O. N.; et al. Ekol. Promyshl. Proizv. 2007, No 4, 45–47. [99] Esis, A. A.; Turik, A. V.; Verbenko, I. A.; et al. Konstr. Kompozits. Mater. 2007, No 1, 82–93. [100] Turik, A. V.; Sidorenko, E. N.; Zhestkov, V. F.; Komarov, V. D. Bul. of Acad. Sci SSSR, Phys. 1970, vol 34, 2590–2593. [101] Akbaeva, G. M.; Gavrilyachenko, V. G.; Kuznetsova, E. M.; et al. In Proc. VIII Int. Interdisciplinary Symp. “Order, Disorder, and Properties of Oxides” (ODRO-2002), Rostov-on-Don, 2002, pp 208–210.

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[102] Turik, A. V.; Sidorenko, E. N.; Zhestkov, V. F.; et al. Rus. Phys. J., 1972, No 10, 122– 124. [103] Smolenskii, G. N. Physics of Ferroelectricity; Nauka: Leningrad, 1985. [104] Sklyarova, E. A.; Reshetnyak, N. V.; Kuznetsova, E. M.; et al. In Proc. VIII Int. Interdisciplinary Symp. “Order, Disorder, and Properties of Oxides” (ODRO-2002), Rostov-on-Don, 2005, pp. 96–100. [105] Burkhanov, A. I.; Alpatov, A. V.; Shil‘nikov, A. V.; et al. Phys. Stat. Sol. (S.Peterburg), 2006, vol 48, 1047–1048. [106] Esis, A. A.; Verbenko, I. A.; Yurasov, Yu. I.; et al. Issl. Ross. 2007, vol 81, 848–855, http://zhurnal.ape.relarn.ru/articles/2007/081.pdf.

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In: Piezoceramic Materials and Devices Editor: Ivan A. Parinov, pp. 71-112

ISBN 978-1-60876-459-4 © 2010 Nova Science Publishers, Inc.

Chapter 2

HIGH PERFORMANCE OF ADVANCED COMPOSITES BASED ON RELAXOR-FERROELECTRIC SINGLE CRYSTALS (1)

V.Yu. Topolov(1),1 C.R. Bowen(2),2 and S.V. Glushanin(3),3

Department of Physics, Southern Federal University4, Rostov-on-Don, Russia: (2) Materials Research Centre, Department of Mechanical Engineering, University of Bath, Bath, United Kingdom (3) Scientific Design & Technology Institute ―Piezop ribor‖, Southern Federal University & Shakhty Institute (Branch), South-Russian State Technical University (NPI)5, Shakhty, Russia

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ABSTRACT The outstanding electromechanical properties of single crystals of relaxorferroelectric solid solutions near the morphotropic phase boundary suggest that these materials can be regarded as potential components of high-performance piezo-active composites. In this chapter we discuss the advantages of using the relaxor-ferroelectric single crystals of solid solutions of (1 – x)Pb(Mg1/3Nb2/3)O3 – xPbTiO3 and (1 – y)Pb. . (Zn1/3Nb2/3)O3 – yPbTiO3 as components of single crystal / polymer and single crystal / porous polymer composites of the 2–2 and 1–3 types. Examples of the high piezoelectric activity and sensitivity and large hydrostatic parameters of the aforementioned composites are analysed in connection with the electromechanical properties of the single-crystal component. The influence of the polarisation orientation on the effective electromechanical properties and related parameters of the single crystal / polymer composite are discussed for different connectivity patterns. Comparisons are made between the effective parameters of 2–2 connectivity composites based on polydomain 1

[email protected]. [email protected]. 3 [email protected]. 4 5 Zorge Street, 344090 Rostov-on-Don, Russia. 5 346500 Shakhty, Rostov Region, Russia. 2

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V.Yu. Topolov, C.R. Bowen and S.V. Glushanin single crystals and single-domain single crystals of the same chemical composition (for instance, 0.67Pb(Mg1/3Nb2/3)O3 – 0.33PbTiO3). Specific effective parameters of single crystal / polymer composites are compared to those of piezo-active composites based on the conventional ferroelectric ceramic and having the same connectivity pattern (either 2–2 or 1–3). In general, data presented in the chapter show how the relaxor-ferroelectric single-crystal component improves the effective parameters of the novel composites and promotes the formation of non-monotonic volume-fraction dependences of the effective parameters which are of particular interest for a variety of piezotechnical applications, such as active elements of sensors, actuators, transducers, hydrophones, etc.

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1. INTRODUCTION Perovskite-type ferroelectric ceramics [1–3] are well-known components of piezo-active composites that are manufactured, studied, and developed in the last decades [4–9]. Trends in manufacturing and research reveal that the new generation of relaxor-ferroelectric single crystals (SCs) with outstanding electromechanical properties [10–13] has significant potential for the further development of the composite branch of modern materials science and enables the development of high performance piezo-composites [15–18] and advanced smart materials [5]. A study on the performance of the piezo-composites based on SCs implies knowledge of the interconnections between the electromechanical properties of components and the ability to take into account the anisotropy of the properties of SC and the microgeometry of the composite [9]. It is remarkable that ferroelectric SCs and ceramics often find applications due to their important piezoelectric properties [3]. Due to the considerable piezoelectric effect and electromechanical coupling in ferroelectric materials [1– 3, 10, 19], a highly effective electromechanical transformation of energy takes place and finds various applications. Piezoelectricity as a physical phenomenon underlining the key role of electromechanical coupling in dielectric solids is observed in acentric SCs including those showing ferroelectric properties [20, 21]. Recent research results in the rapidly growing field of smart materials [2, 9, 19] demonstrate that both ferroelectricity and piezoelectricity represent an important link between solid-state science and engineering. It is obvious that knowledge of the electromechanical properties and possibilities of control over the properties are necessary conditions for the correct use of SCs as piezo-active components of advanced composites [9, 15–18]. It is important to note that variations in microgeometry (connectivity) of the composite, volume fraction, chemical composition, and domain structure of SCs lead to changes in the physical properties and their anisotropy [9, 15–18]. The aim of this chapter is to consider the effective electromechanical properties and advantages of the piezo-composites that are based on relaxor-ferroelectric SCs and are useful in a variety of piezotechnical applications.

2. RELAXOR-FERROELECTRIC SOLID SOLUTIONS AND THEIR ELECTROMECHANICAL PROPERTIES Relaxor-ferroelectric perovskite-type solid solutions are materials with the general formula (1 – x)Pb(B1, B2)O3 – xPbTiO3. The complex perovskites Pb(B1, B2)O3 are

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disordered compounds that contain ions of metals from the following groups: B1 = Mg, Zn, Ni, Fe, Sc, Yb, and In (low valence) and B2 = Nb, Ta and W (high valence) [10, 12, 22]. The combination of metal ions with low and high valences results in physical properties that distinguish Pb(B1, B2)O3 from ― normal‖ (ordered or regular) perovskite-type ferroelectrics such as PbTiO3, BaTiO3 or KNbO3. The complex perovskites Pb(B1, B2)O3 exhibit a broad and frequency-dispersive dielectric maxima and contain polar nanoregions (with ferro- or antiferroelectric ordering) in a non-polar phase over a wide temperature range [10, 22]. These materials are characterised by the relaxation dielectric polarisation and are called relaxors or ferroelectric relaxors [10, 22]. The solid solutions of (1 – x)Pb(B1, B2)O3 – xPbTiO3 combine physical properties of the relaxor-type and ― normal‖ ferroelectric components, and, as a rule, excellent electromechanical properties are observed near the morphotropic phase boundary (MPB) [11– 14]. To achieve a high piezoelectric activity in SCs, the relaxor-ferroelectric solid solutions are engineered by compositional adjustment with a corresponding decrease in Curie temperature of the paraelectric-to-ferroelectric phase transition [11, 12]. In this connection important experimental work was performed to obtain high-performance domain-engineered or heterophase SCs of relaxor-ferroelectric solid solutions of the (1 – x)Pb(B1, B2)O3 – xPbTiO3 type. Among these compounds, of practical interest are domain-engineered SCs of (1 – x)Pb(Mg1/3Nb2/3)O3 – xPbTiO3 (PMN–xPT) and (1 – y)Pb(Zn1/3Nb2/3)O3 – yPbTiO3 (PZN–yPT) with molar concentrations x and y, respectively, taken in the vicinity of the MPB. Engineered non-180 domain structures [11, 14, 23, 24], intermediate ferroelectric phases [13, 14, 25–28] and domain-orientation processes [13, 14, 29, 30] play an important role in forming the outstanding electromechanical properties of these materials. Due to these and other phenomena, PMN–xPT and PZN–yPT SCs poled along certain crystallographic directions (often along [001] or [011] of the perovskite unit cell) exhibit very high piezoelectric activity and significant electromechanical coupling [31–40]. Examples of full sets of electromechanical constants of domain engineered PMN–xPT and PZN–yPT SCs near the MPB are shown in Figure 1. We see that the piezoelectric coefficients d3(1j)  103 pC / N for various SC compositions remains larger than d3(1j) of conventional ferroelectric ceramics based on Pb(Zr1–zTiz)O3*) [1–3, 21, 22]. Moreover, according to data [41], the piezoelectric (1) coefficient d33 in [001]-poled heterophase PZN–0.08PT SCs can attain values up to 12 000

pC / N due to the electric-field-induced phase transition and the presence of the intermediate monoclinic phase. However, in the single-domain state of SCs and in case of the poled ceramic composition (see Figures 2 and 3), the typical values are only d3(1j)  102 pC / N [36, 39, 42]. As follows from experimental research [11, 12], relatively high piezoelectric strains (over 0.5 %) are achieved in the [001]-poled domain engineered SCs of relaxor-ferroelectric solid solutions under an electric field E || [001]. Such SCs are described by the 3m point group in the single-domain state, and the spontaneous polarisation vectors of domains in domain

*

Hereafter we use superscript (1) to denote the electromechanical properties of the SC component in the composites being considered.

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V.Yu. Topolov, C.R. Bowen and S.V. Glushanin

engineered samples are parallel to the [111], [ 1 11], [1 1 1], and [ 1 1 1] directions [23, 43]. Important interconnections between the structure and properties in the PMN–xPT and PZN– yPT solid solutions have been reviewed [44]. The high piezoelectric activity in these solid solutions can be associated with a polarisation rotation induced by the electric field E [13, 29]. This rotation between the single-domain states in the tetragonal (4mm) and rhombohedral (3m) phases of the ferroelectric nature can be implemented in different ways that form intermediate monoclinic phases and complicated heterophase states [29, 44–46]. The polarisation rotation paths found [13] are the minimum free-energy paths and are associated with intermediate ferroelectric phases that would be observed in certain ranges of molar concentration x or y, electric field E and temperature T.

Figure 1. Experimental values of elastic compliances

(1), E (in 10-12 Pa-1), piezoelectric coefficients sab

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dij(1) (in pC / N) and dielectric permittivities  (pp1), / 0 of domain engineered PMN–xPT and PZN–yPT SCs near the MPB at room temperature.

Figure 2. Experimental values of elastic compliances

(1), E (in 10-12 Pa-1), piezoelectric coefficients sab

dij(1) (in pC / N) and dielectric permittivities  (pp1), / 0 of single-domain PMN–xPT SCs (4mm symmetry) near the MPB at room temperature.

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Figure 3. Experimental values of elastic compliances

75

(1), E (in 10-12 Pa-1), piezoelectric coefficients sab

dij(1) (in pC / N) and dielectric permittivities  (pp1), / 0 of the poled PMN–0.35PT ceramic at room temperature [42].

The outstanding electromechanical properties of PMN–xPT and PZN–yPT SCs near the MPB (Figure 1) suggest that these materials are excellent candidates for modern piezoelectric sensors, large-strain actuators, highly sensitive medical ultrasonic transducers, hydrophones, ultrasonic imaging devices, and other piezotechnical devices. Below we show possibilities and advantages of using these SCs as piezo-active components of high-performance composites. We demonstrate how PMN–xPT and PZN–yPT SCs improve the performance of the piezo-composites, as compared with the performance of the conventional ferroelectric ceramic / polymer composites, and promote the formation of important volume-fraction dependences of the effective electromechanical properties and other parameters.

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3. COMPOSITES WITH 2–2-TYPE CONNECTIVITY AND THEIR PERFORMANCE The composite with 2–2 connectivity (in terms of classification proposed by Newnham et al. [4, 5], hereafter referred to as the 2–2 composite) represents a system of layers of two components. As a rule, the layers are lengthy in two co-ordinate directions and separated by parallel interfaces. The family of the 2–2 piezo-composites comprises laminar structures with the regular or irregular distribution of components, and with different poling directions and orientations of the interfaces with respect to the poling direction (see, e.g. [4–6]). According to data [47], the 2–2 connectivity pattern is one of the so-called junction connectivity patterns and their combination can form new connectivity patterns of the twocomponent composites with planar microgeometry. In this section we consider examples of the performance of the relaxor-ferroelectric SC-based composites of the 2–2 type and orientation effects in these composites.

3.1 2–2 Parallel-Connected Composite Based on PZN–0.07PT: Effect of the Orientation of Crystallographic Axes As follows from experimental data, full sets of electromechanical constants have first been measured on PZN–0.07PT SCs poled in two different directions, namely [001] and [011], with respect to the perovskite unit-cell axes [32, 34]. Data from Figure 4 suggest that both symmetry and anisotropy differ from those peculiar to the domain engineered [001] poled PZN–xPT or PMN–yPT SCs (see Figure 1) and conventional poled ferroelectric ceramics [1, 3, 21]. For example, in the [011] poled PZN–0.07PT SC elastic compliances (1), E (1), E (1), E (1), E (1), E (1), E are linked by the following ratios: s11 / s12 = –1.12, s11 / s13 = 20.1 and s11 / sab

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V.Yu. Topolov, C.R. Bowen and S.V. Glushanin

(1), E = 1.09. The piezoelectric coefficients d3(1j) of the same [011] poled SC are characterised s33

[34, 48] by ratios (1) (1) (1) (1) / d 31 = 2.41 and d 33 / d 32 = –0.788, d 33

(1)

(1) (1) (1) (1) that is, condition d 33 / d 31 = d 33 / d 32 holding true in the [001] poled SCs (see data from

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Figure 1) and ferroelectric ceramics [1, 20, 21] is violated in case of [011] poling. As shown recently [48], Eq. (1) and other features of the electromechanical properties of this SC play an important role in forming the volume-fraction and orientation dependences of the effective properties of the 2–2 PZN–0.07PT SC / polymer composite.

Figure 4. Experimental values of elastic compliances

(1), E (in 10-12 Pa-1), piezoelectric coefficients sab

dij(1) (in pC / N) and dielectric permittivities  (pp1), / 0 of domain engineered PZN–0.07PT SCs at room temperature. All constants listed in the chart have been determined in the main crystallographic axes. The [001] and [011] poled SCs are characterised by 4mm and mm2 symmetry, respectively, but in the single-domain state the SC belongs to 3m symmetry class.

The 2–2 composite (Figure 5) contains a system of parallel-connected layers which are made from SC and polymer and alternating in the OX1 direction. The main crystallographic axes X, Y and Z in each SC layer are parallel to the following perovskite unit-cell directions [34]: X || [0 1 1], Y || [100] and Z || [011]. The SC layers represent cuts concerned with a clockwise rotation of the main crystallographic axes X and Y around the co-ordinate axis OX3 by the rotation angle . The angle  = 0 means that conditions X || OX1, Y || OX2 and Z || OX3 hold. The composite sample as a whole is poled along the OX3 axis (Figure 5). An averaging procedure for a determination of the effective electromechanical properties of the 2–2 SC / polymer composite is implemented by taking into account the nine boundary conditions [9, 48] at x1 = const. These boundary conditions require the continuity of three normal components of the mechanical stress (11, 12 and 13), three tangential components of the mechanical strain (22, 23 and 33), one normal component of the electric displacement

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(D1), and two tangential components of the electric field (E2 and E3). The effective electromechanical properties of the 2–2-type composite are determined using a matrix [48, 49] || C* || = (|| C(1) |||| M || m + || C(2) || (1 – m))(|| M || m + || I || (1 – m))-1.

(2)

In Eq. (2) || C(1) || and || C(2) || are the matrices of electromechanical constants of the SC [superscript (1)] and polymer [superscript (2)] components, || I || is the identity matrix, || M || is the matrix [49] that is concerned with the aforementioned boundary conditions, and m is the volume fraction of SC (0 < m < 1). The || C(n) || from Eq. (2) matrices are given by

 || s ( n ), E || || d ( n ) ||t    ( n ),   , || C(n)|| =  (n) ||   || d || || 

(3)

where || s(n),E || is the 6  6 matrix of elastic compliances (at electric field E = const), || d(n) || is the 3  6 matrix of piezoelectric coefficients and || (n), || is the 3  3 matrix of dielectric permittivities (at mechanical stress  = const) of the SC (n = 1) and polymer (n = 2) components, and superscript t denotes the transposition. The || C* || matrix from Eq. (2) has a form similar to that shown in Eq. (2). The effective electromechanical properties of the composite, i.e., full sets of

* * *E sab (m, ), dij (m, ) and  pp (m, ), are determined within the

framework of the so-called longwave approximation [9, 49]. This approximation means that the wavelength of acoustic waves propagated is considerably longer than the thickness of each layer (Figure 5) in the composite structure. Based on elements of || C*(m, ) || from Eq. Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

(2), we evaluate the effective piezoelectric coefficients

eij* , gij* ,

and

hij* ,

the hydrostatic

piezoelectric coefficients

dh*

=

* d33

+

gh*

=

* g33

+

* d32

+

* d31

(4)

and * * * * g32 + g31 = dh /  33 ,

(5)

squared figures of merit * * 2 * * * = ( d33 )2 /  33 (Q33 ) = d33 g33

(6)

* , (Qh* ) 2 = dh* gh* = ( dh* )2 /  33

(7)

and

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V.Yu. Topolov, C.R. Bowen and S.V. Glushanin

and other parameters of the 2–2 composite. The piezoelectric coefficients from Eqs. (4) and (5) are used to describe hydrostatic activity and sensitivity, respectively. Squared figures of merit from Eqs. (6) and (7) are introduced [5, 9, 50] to describe the sensor signal-to-noise ratio of the piezoelectric element.

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Figure 5. Schematic of the 2–2-type composite with parallel-connected layers. (X1X2 X3) is the rectangular co-ordinate system. m and 1 – m are volume fractions of the SC and polymer components, respectively, and  is the angle of rotation of the main crystallographic axes X and Y of the SC around the OX3 axis where OX3 || Z (reprinted from Krivoruchko and Topolov [48], with permission from IOP Publishing).

Data on local maxima of effective parameters (*)m = max *(m, ) | = const of the 2–2 SC / polymer composites are shown in Figure 6. Both the piezoelectric anisotropy (see Eq. (1)) and the elastic anisotropy of the SC component influence the orientation dependence of a series of the effective parameters [48]. The effective parameters

* * * 2 g33 , gh , (Q33 ) ,

and

(Qh* ) 2 , which are concerned with the piezoelectric sensitivity of the composite, increase when araldite is replaced with a softer polymer. The presence of the very soft polymer (i.e., elastomer) in the 2–2 composite leads to the large hydrostatic piezoelectric activity (see data from Figure 6 and curves 1–4 in Figure 7, a). Examples of non-monotonic volume-fraction behaviour shown in figures. 6 and 7 are connected with distinctions between the properties of the SC and polymer components, as well as with a considerable redistribution of effective and mechanical fields in the composite sample with interfaces x1 = const (Figure 5). Data from Figure 6 show that a considerable increase in the piezoelectric sensitivity of *

the 2–2 composite is predicted at the rotation angle   90, i.e. ( g33 )m / *

( gh )m /

(1) g 33 

63 and

g h(1)  75. At the same time the anisotropy of elastic properties of the SC and an

increase in softness of the polymer component promote an increase in the piezoelectric coefficients

* h33

and

* e33 , and the following ratios are attained in the 2–2 PZN–0.07PT SC /

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elastomer composite: ( h33 )m /

(1) h33

*

= 2.9 and ( e33 )m /

79

(1) e33 = 4.7 [48]. In 2–2 and other two-

component piezo-composites research, these unusually large ratios related to the piezoelectric coefficients

* h33

and

* e33 are first mentioned in [48]. For example, the 1– PZT-type ceramic

*

/ elastomer composites with  = 1, 2 or 3 are characterised by max e33 / where

* e33

( FC ) e33

is a function of the volume fraction of the ceramic component and

= 1.16 [47], ( FC ) e33

is its

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piezoelectric coefficient.

Figure 6. Local maximum values of the following effective parameters calculated for 2–2 PZN–0.07PT * SC / polymer composites in ranges of 0 < m < 1 and 0  θ  90: Piezoelectric coefficients ( g33 )m (in * * mV.m / N), ( h33 )m (in 108 V / m), ( e33 )m (in С / m2), hydrostatic piezoelectric coefficients ( dh* )m (in

pС / N), ( gh* )m (in mV.m / N) and squared figures of merit Рa-1). Dash denotes monotonic increasing

dh* (m, )

* 2 (Q33 )m (in 10-12 Рa-1) and (Qh* )2m (in 10-15

at  = const (reprinted from Krivoruchko and

Topolov [48], with permission from IOP Publishing).

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V.Yu. Topolov, C.R. Bowen and S.V. Glushanin

Figure 7. Effective parameters calculated for the 2–2 PZN–0.07PT SC / elastomer composite: (a) hydrostatic piezoelectric coefficients dh* (in pC / N) and gh* (in mV.m / N) and (b) squared hydrostatic figure of merit

(Qh* ) 2 (in 10-12 Рa-1) (reprinted from Krivoruchko and Topolov [48], with permission

from IOP Publishing).

The high piezoelectric activity of the 2–2 composite at volume fractions of SC m < 0.10 (curves 1–4 in Figure 7, a) favours high piezoelectric sensitivity (curves 5–8 in Figure 7, a

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High Performance of Advanced Composites… and curves 1–4 in Figure 7, b). The hydrostatic parameters

81

gh* and (Qh* ) 2 from ranges 0.01 < *

m < 0.10 and 45 <  < 90 (Figure 7) are approximately 5 – 6 times larger than max gh and * 2

max[ (Qh ) ], which were earlier predicted [51] for the 1–3 PZN–xPT SC / araldite composite (x = 0.045, 0.07 and 0.08) with a system of SC rods poled along [001].

3.2

2–2–0 Composite Based on PZN–0.07PT: Effect of Porosity in Polymer Layers In this section we analyse the influence of porosity on the effective electromechanical properties of the studied 2–2-type composite based on the PZN–0.07PT SC. It is assumed that each polymer layer of the composite (see Figure 5) contains a system of isolated pores in a polymer matrix (3–0 connectivity). These pores in the whole composite sample can be either spherical and distributed randomly (P-1 matrix) or in the form of oblate spheroids and distributed regularly. The spheroidal pores are described by equation in the (X1X2X3) system: (x1 / a1)2 + (x2 / a2)2 + (x3 / a3)2 = 1.

(8)

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Semi-axes ai of spheroids from Eq. (8) are linked by aspect ratios ρ1 = a3 / a1 = a3 / a2 < 1 (P-2 matrix) or ρ2 = a1 / a2 = a1 / a3 < 1 (P-3 matrix). The corresponding composites with the porous P-k matrix (k = 1, 2 or 3) in each polymer layer are described by 2–2–0 connectivity. The air pores are characterised by the volume fraction mp in the polymer matrix, and sizes of these pores are considerably less than a thickness of the polymer layer on the OX1 direction. The elastic and dielectric constants of the piezo-passive porous polymer with 3–0 connectivity are determined using formulae [52]. Data from work [48] suggest that the influence of porosity in the polymer layers on the hydrostatic parameters of the 2–2–0 composite [see Eqs. (4), (5) and (7)] is more significant in case of the PM-2 matrix. The arrangement of the oblate pores described by Eq. (8), as well as their shape (1 < 1) and changes in porosity mp, result in the specific ratios of the elastic compliances

( 2) s11

/

( 2) s13

and

( 2) s11

/

( 2) s33

of the porous polymer layer. The presence of the

porous P-k matrix leads to a considerable increase in local maxima of the hydrostatic piezoelectric coefficient

gh* (curves 2, 4 and 6 in Figure 8, a) and squared figures of merit

* 2 (Q33 ) and (Qh* ) 2 (Figure 8b), as compared with the similar parameters of the related 2–2

composite (see Figures 6 and 7). The porous structure influences a balance of the piezoelectric coefficients

* 2 d3* j that promote a change in relations between (Q33 ) and (Qh* ) 2

(see Eqs. (6) and (7) and compare curves 1, 2 and 3 in Figures 8, b and c). It should be noted *

2

for comparison that the local maximum values of [ (Q33 ) ]m  10-9 Pa-1 (Figures 8, b and d) *

2

and max (Q33 )  10-9 Pa-1 in the 1–3–0 PZT-type ceramic / porous elastomer composite [53] are of the equal order-of-magnitude. Data in Figures 8, a, c and e suggest that the largest values of the hydrostatic parameters are attained in the 2–2–0 composites with the porous P-k matrices in the vicinity of  = 90. The corresponding orientation of the main crystallographic axes of the SC (Figure 5) weakens an influence of the piezoelectric coefficient of SC

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(1) 0 on the hydrostatic d31

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Figure 8 (Continued).

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V.Yu. Topolov, C.R. Bowen and S.V. Glushanin

Figure 8. Effective parameters of the 2–2–0 PZN–0.07PT SC / porous elastomer composites with different porous P-k matrices (k = 1, 2 or 3) at porosity mp = 0.2: (a) local maxima of hydrostatic piezoelectric coefficients ( dh* )m (in pC / N) and ( gh* )m (in mV.m / N), (b) local maxima of squared *

* 2

2

figure of merit [ (Q33 ) ]m (in 10-11 Pa-1), (c) local maxima of squared figure of merit [ (Qh ) ]m (in 10-12 Pa-1), (d) squared figure of merit

* 2 (Q33 )

(e) hydrostatic squared figure of merit

(in 10-11 Pa-1) of the composite with the P-2 matrix, and

(Qh* )2

(in 10-12 Pa-1) of the composite with the P-2 matrix

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(reprinted from Krivoruchko and Topolov [48], with permission from IOP Publishing).

The increase of

(Qh* ) 2 at 45 <  < 90 (Figure 8, e) is also accounted for by the

influence of the piezoelectric coefficients

(1) (1) and d32 on the hydrostatic response of the 2– d31

* 2

2–0 composite. It is obvious that values of max[ (Qh ) ] for 45 <  < 90 are attained at volume fractions m* < 0.02 (Figure 8, e). Increasing the volume fraction m by about 0.05 (in comparison to the volume fraction m* for   60, see curves 2 – 4 in Figure 8, e) does not lead to drastic decreasing

(Qh* ) 2 : this parameter remains about 3 – 5 times larger [48] than * 2

the experimental value of max[ (Qh ) ] = 1.4.10-11 Pa-1. This value has been found for the 2–2 oriented PZT FC / polymer composite manufactured and studied in work [6]. The values of the hydrostatic piezoelectric coefficient

dh* calculated for the 2–2–0 composite at 0.1  mp 

0.2 and various volume fractions m (see, for instance, curves 1, 3 and 5 in Figure 8, a) can *

exceed max dh = 194 pC / N for the 1–3 PZN–0.07PT SC / araldite composite [53] with the SC component poled along the [001] direction.

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3.3 2–2 Composite Based on PZN–0.07PT: Illustrative Example of Combination of Electromechanical Properties Three main effects [9, 54] that contribute to the effective properties of composites based on ferroelectrics are (1) the effect of the addition of the properties, (2) the effect of the combination of the properties, and (3) the effect of the generation of the properties. An interesting example of the combination property in the 2–2 PZN–0.07PT SC / polyvinylidene * * fluoride composite [55] concerns the piezoelectric coefficient h33 = h33 (m, θ). The study of

the effective piezoelectric properties has been carried out [55] within the framework of the model of the parallel-connected 2–2 composite (Figure 5). Polyvinylidene fluoride (PVDF) is the polymer component that exhibits ferroelectric and piezoelectric properties [56]. The components in the studied composite provide unique equalities of the piezoelectric (1) ( 2) ≈ – h33 (when the spontaneous polarisation vector of SC Ps(1)  OX3 and coefficients: h33 (1) ( 2) the remanent polarisation vector of polymer Pr(2)  OX3) and h33 ≈ h33 (when Ps(1) 

OX3 and Pr(2)  OX3). However, these distinctions in the piezoelectric properties of the SC * and polymer components influence values of max h33 slightly (Figure 9). According to data * (1) * (1) from work [55], max h33 = 2.85 h33 and max h33 = 2.88 h33 would be attained in the

composite with the polymer layers having Pr(2)  OX3 and Pr(2)  OX3, respectively. The * values of max h33 are attained at volume fractions of SC m ≈ 0.5 – 0.6, and therefore, the * effect of dielectric properties of the polymer component on h33 vanishes. It has been shown

* * *D * *D * *D = g31 + g32 + g33 remains almost [55] that the piezoelectric coefficient h33 c13 c23 c33

constant due to a simultaneous monotonic decrease in the absolute values of the piezoelectric coefficients | g 3* j | and monotonic increase of the elastic moduli c*j 3D . The composite structure Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

(Figure 5) and the combination of the piezoelectric and elastic properties favour the * fulfillment of the condition h33 (m, θ) ≈ const. It is noteworthy that maximum values of

* (n) (m, θ) (Figure 9) are approximately three times more than | h33 |, and the behavior of such h33

piezoelectric coefficients has not been previously observed [55].

3.4

2–2 Composite Based on PMN–0.33PT: Key Role of the Single-Domain State PMN–0.33PT is the only relaxor-ferroelectric SC for which full sets of electromechanical constants were measured in both the single-domain [57] and polydomain (domain engineered) [31] states at room temperature. In the single-domain state (3m symmetry), the electromechanical properties were determined in the main crystallographic axes (Figure 10). In this section we compare the effective piezoelectric properties of the 2–2 parallel-connected SC / polymer composites based on either the single-domain or polydomain SC component.

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Figure 9. Effective piezoelectric coefficient

* h33

(in 108 V / m) calculated for the 2–2 [011] poled

PZN–0.07PT SC / PVDF composite with PVDF-1 (a), PVDF-2 (b), or PVDF-3 (c) layers. Dashed lines are guides for eyes (reprinted from Topolov and Krivoruchko [55], with permission from the American Institute of Physics).

Figure 10. Experimental values of elastic compliances

dij(1)

(in pC / N) and dielectric permittivities

 (pp1),

(1), E (in 10-12 Pa-1), piezoelectric coefficients sab

of single-domain PMN–0.33PT SC (3m symmetry)

at room temperature [57].

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It is assumed that there is a regular distribution of layers in the OX1 direction, and the layers are continuous in the OX2 and OX3 directions (Figure 11). In this section we consider composite-1 as a material that comprises the single-domain SC and polymer layers and composite-2 as a related material based on the polydomain SC with the same chemical composition (cf. insets 1 and 2 in Figure 11). The orientation of the spontaneous polarisation vector Ps(1) of each SC layer in composite-1 is characterised by Euler angles θ, ψ and θ (inset 1 in Figure 11). In composite-2, the spontaneous polarisation vector Ps(1) of each polydomain SC obeys condition Ps(1)  ОХ3 (inset 2 in Figure 11). The layers of ferroelectric PVDF in composite-1 and composite-2 have the remanent polarisation vector oriented as follows: either Pr(2)  ОХ3 (PVDF-1 in the composite sample) or Pr(2)  ОХ3 (PVDF-2 in the composite sample) (see insets 3 and 4 in Figure 11). The manufacture of such a 2–2 composite with components having various orientations of the Ps(1) and Pr(2) vectors can be achieved by a consequent poling of these components. In the case of the PMN–0.33PT SC / (n )

of SC (n = 1) and polymer (n = 2) obey the condition

PVDF composite, coercive fields Ec (1)

( 2)

[34, 56] Ec D) to stochastic type (λ ~ D) and scattering character from multiple scattering on randomly (radial mode) and regularly distributed particles (thickness mode). Scattering on micro pores is negligible in this frequency range. Frequency dependencies of real parts of elastic moduli С’33D and corresponding loss tanδ = С’’33D /С’33D for different volume fractions of scattering particles measured by impedance spectroscopy method are shown in Figure 14. It is readily seen that the elastic moduli С’33D increase with frequency because of the frequency dispersion caused by the scattering of ultrasonic waves. It also can be seen that the frequency dependencies of the scattering loss tgδ are linear and correspond to stochastic scattering type tanδ ~ Df at λ ~ D. Figure 15 shows the dependencies of piezomoduli d33, measured by a quasi-static method, and d31, measured on radial mode, as well as electromechanical coupling factors for thickness kt and radial kp modes on volume fraction of scattering phase. It is readily seen in Figure 15 that a decrease in d33quasi follows the simple law d*33 = d33quasi (dense)  d33quasi (dense) ∙ S, where S = V2/3 is surface fraction of non-piezoelectric particles, d33quasi (dense) – quasistatic piezomodule at V% = 0.

Figure 13. Elastic moduli С’33D, С’33E, S’11E and corresponding mechanical quality factors QM = S’11E / S’’11E, С’33E /С’’33E and С’33D /С’’33D versus volume fraction of scattering particles V% in PZT matrix.

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Figure 14. Frequency dependencies of elastic moduli С’33D and corresponding loss tanδ = С’’33D / С’33D for different volume fractions V% of the scattering particles in PZT matrix.

Figure 15. Piezoelectric moduli d33, d31, thickness kt and planar kp, electromechanical coupling factors versus volume fraction of scattering particles V% in PZT matrix.

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Dynamically measured, d31 drops with V% faster than d*31, as a result of the inequality of composite structure in radial and thickness directions, above mentioned. Thickness mode coupling factor kt shows minor changes in all measured V% range, despite growth of nonpiezoelectric phase content. This is caused by the relative porosity growth and formation of quasi-rod structure of piezoceramic matrix [2, 5]. Planar coupling factor kp decreases with V% drastically as a combined result of non-piezoelectric phase content growth and disintegration of composite connectivity in radial direction. We can summarize the preceding frequency and volume fraction dependencies of the main composite parameters as follows: optimal materials for extreme damping at reasonable electromechanical parameters are ceramic composites with 15 ÷ 25 vol.% of α-Al2O3 particles in soft PZT matrix at λ/D ratio ~ 10. In conclusion, the advantages of ceramic composites are demonstrated with reference to implemented ultrasonic transducers. Figure 16 shows pulse echo and transmit-mode characteristics (hydrophone signal) of PZT/α-Al2O3 elements and ultrasonic transducer loaded on water and Perspex. Piezoelements were excited by spike pulses. Echo signals in water were received from Perspex reflector. Figure 16 also shows that a low-Q and low-ZA composite element demonstrates highdamped and high amplitude signals even without matching and damper. Notice also that the transducer made of new ceramic composites exhibits a smaller ripple, a larger bandwidth and is free of harmonics, i.e., the spatial resolution is higher and its dynamic range broader. Keeping the optimal scatterers size-to-wavelength ratio, we manufactured and tested a line of wide-band NDT ultrasonic transducers made from new ceramic composites with high sensitivity and resolution in frequency range 0.2 ÷ 10 MHz.

A Figure 16 (Continued).

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b Figure 16. Transmit (a) and pulse-echo characteristics with corresponding FFT (b) for A850L-20 composite elements.

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A new family of low-Q ceramic piezocomposites was developed based on the original microstructural design concept (MSD) using ― damping by scattering‖ approach. The main advantages of the new PZT/α-Al2O3 piezocomposites are the result of high acoustic efficiency, low crosstalk, and low mechanical Q. The gain will be especially in high sensitivity combined with well-damped signals. Additional advantages of the developed piezocomposites possible with controllable changes of main properties in a wide range, compatibility with standard fabrication technologies and processing flexibility.

7. DIELECTRIC, PIEZOELECTRIC AND ELASTIC PROPERTIES OF PZT/PZT CERAMIC PIEZOCOMPOSITES A new method of low-Q PZT/PZT ceramic piezocomposites fabrication is proposed in [70, 71]. Different types of PZT type piezoceramics powders and milled PZT piezoceramic particles, as well as pre-sintered piezoceramic granules were used as matrix and filling components, respectively. Samples of piezocomposites with the volume fraction of components 0-100% were manufactured and tested. Complex sets of elastic, dielectric, and piezoelectric parameters of piezocomposites were measured by impedance spectroscopy method using piezoelectric resonance analysis (PRAP) software. Velocity and attenuation of longitudinal and shear waves for polarized and non-polarized samples were measured by pulse-echo and through-transmit ultrasonic methods.

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7.1. Experimental Procedures At first, a line of chemically, thermally, and technologically compatible ceramic matrix and scattering phase materials were chosen. Different types of raw PZT powders and milled PZT piezoceramic particles as well as pre-sintered PZT granules were used as initial components for ceramic composite preparations. Special pressing and firing regimes and porosifiers were used for the formation of microporous piezoceramic matrices. Sintering of the green bodies was carried out at special thermal profiles to prevent cracking caused by difference in shrinkage and thermal expansion coefficients of the composite components. Figure17 shows two examples of ceramic composite microstructures.

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a)

b) Figure 17. SEM micrographs of ceramic composite structures: (a) pre-sintered PZT granules in porous PZT matrix; (b) milled dense PZT particles in porous PZT matrix.

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7.2. Methods of Measurement Complex electrical coefficients of piezocomposite elements were determined by impedance spectroscopy using Piezoelectric Resonance Analysis (PRAP) software [21, 68]. Measurements were made on Solartron Impedance/Gain-Phase Analyzer SL 1260. Sound velocity and attenuation of longitudinal and shear waves in different directions of polarized and non-polarized samples were measured by pulse–echo and through-transmit ultrasonic methods in the frequency range 1 ÷ 5 MHz using digital LeCroy oscilloscope, Olympus pulser/receivers and various Olympus transducers. Microstructures of polished, chemically etched, and shipped surfaces of composite samples were observed with scanning electron microscopes (SEM, Karl Zeiss).

7.3. Results and Discussion

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PZT/PZT ceramic piezocomposites composed by the hard PZT matrix with randomly distributed pre-sintered PZT granules with a mean particle diameter ~ 30 μm and volume fraction from 0 up to 100 m% were chosen as model samples for illustration of the ― damping by scattering‖ approach. Figure 18 shows shrinkage coefficient Kshdiam for cylindrical composite elements Ø23  20 mm2 as a function of concentration of pre-sintered ceramic granules m% at the same sintering regime. It is readily seen in Figure 18 that Kshdiam decreases drastically with m% caused by the increase of the nonshrinking phase concentration (pre-sintered ceramic granules), which prevents shrinkage of ceramics matrix and leads to microporosity appearance.

Figure 18. Dependence of shrinkage coefficient Kshdiam. of composite elements on concentration of presintered ceramic granules m%.

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The dependencies of theoretical ρtheor and measured ρexper density, as well as relative porosity P% of ceramic composite on concentration of pre-sintered ceramic granules m% are shown on Figure 19.

Figure 19. Theoretical ρtheor and measured ρexper density, as well as relative porosity P% of ceramic composite as function of concentration of pre-sintered ceramic granules m%.

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It is readily observed in Figure 19 that the density of the ceramic composite drops and relative porosity grows rapidly with concentration m%, which corresponds well to shrinkage coefficient behavior (Figure 1). Figure 20 shows piezoelectric modulus d33, d31 and relative dielectric constant ε33T/ε0 of ceramic composites as function of concentration of pre-sintered ceramic granules m%.

Figure 20. Dependences of piezoelectric modulus d33, d31, and relative dielectric constant ε33T/ε0 on concentration of pre-sintered ceramic granules m% for ceramic composites PZT/PZT.

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The dielectric constant of the composite 33T/0 decreases drastically with m% because porosity grows (i.e., the dielectric constant of air is much less than for PZT ceramics). The piezoelectric modulus d33 for ceramic composite has minor changes in all m% range caused by continuity of rigid ― quasi-rod‖ ceramic skeleton in the polarization direction (sample thickness). Reduction in relative area of the piezoceramic phase in this case is compensated by an increase in relative pressure applied to the ceramic skeleton. Reduction in ‫׀‬d31‫ ׀‬with m% is obvious and is caused by alteration of quasi-rod ceramic skeleton continuity in a lateral direction (i.e., the lateral size of elements usually twenty times more than the thickness). Figure 21 shows electromechanical coupling factors for thickness kt , longitudinal k31 and radial kp vibration modes of ceramic composites as function of the concentration of presintered ceramic granules m%. The behaviour of electromechanical coupling factor kp and k31 of ceramic composites with concentration of pre-sintered ceramic granules m% is determined by a decrease of piezomodulus d31 and dielectric permittivity 33T and a competing increase of elastic compliances S11E and S12E according to following relation: kp2 = 2d312/[33T(S11E + S12E)] and k312 = d312/(33TS11E). The main reason for the decrease of kp and k31 is mentioned above alteration of piezoceramics skeleton continuity in a lateral direction and, as the consequence, an increase in corresponding elastic compliances of porous ceramics.

Figure 21. Dependencies of electromechanical coupling factors kt , k31 and kp on concentration of presintered ceramic granules m% for ceramic composites PZT/PZT.

The electromechanical coupling factor kt slightly increases with m% due to partial removal of mechanical clamping of a porous piezoceramic structure in lateral direction (electromechanical coupling factor for piezoceramic rods equal to k33 , and mechanical clamping of porous piezoceramic skeleton by air is negligible). At a further increase in m%, the electromechanical coupling factor kt decreases slightly because of a reduction in e33 piezoconstant (kt2 = e332/(33SC33D)). The interrelationship of resulting values of considered

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electromechanical coupling factors at any m% is satisfactorily described by the following approximated formula: kt2  (k332 + kp2)/(1  kp2). The dependencies of elastic moduli С33D, С33E and elastic compliances S11E, S11E on the concentration of pre-sintered ceramic granules m% for ceramic composites PZT/PZT are shown on Figure 22. Elastic compliances S11E and S11D of ceramic composite increase with concentration m% because of porosity growth (stiffness decreasing). S11E and S11D increase approximately 4 ÷ 5 times in a concentration range of 0 ÷ 100%. At the same time, elastic moduli С33D and С33E decrease very fast with concentration m% and practically linearly up to m = 60 %. With further m% growth, С33D and С33E decrease slowly, as a result of a rearrangement of porous composite structure and formation of quasi-rod structure in thickness direction. The average decrease in С33D and С33E in concentration range 0 ÷ 100 % is to 7 ÷ 9 times. The difference in C and S behavior is caused by an inequality in porous composite structure in thickness and lateral direction (the lateral size of standard piezoceramic elements usually twenty times more than the thickness). Figure 23 shows velocities of thickness VtD and extended length of V1E modes of ceramic composite disks as a function of pre-sintered ceramic granules concentration m%. Concentration dependence of longitudinal velocity V11E = (1/ρS11E)1/2 is controlled by competing behaviour of density ρ and elastic compliance S11E (S11E increases because of reduction of elastic stiffness of a porous ceramic skeleton). Thickness extensional mode velocity Vt = (C33D/ρ) 1/2 decrease with m%, which is also caused by faster reduction of elastic stiffness of a porous ceramic skeleton (decrease of C33D) compared to decreasing in density ρ.

Figure 22. Elastic moduli С33D, С33E and elastic compliances S11E, S11D as a function of concentration of pre-sintered ceramic granules m%.

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Figure 23. Dependencies of thickness VtD and length extensional V1E mode velocities of ceramic composite disks on concentration m%.

Figure 24 shows mechanical quality factors for thickness QMt and radial QMr vibration modes of ceramic composite disks as functions of pre-sintered ceramic granules concentration m%. Corresponding dependencies for regular porous ceramics are also shown for comparison. QMt and QMr for porous piezoceramics decrease with m% virtually linearly (concentration range m = 0 ÷ 100 % corresponds to porosity range 0 ÷ 30 %), whereas for ceramic composite, concentration dependencies of QMt and QMr are strongly nonlinear and more pronounced. The character of QM dependences on m% for ceramic composites is determined by summary effect of ultrasonic waves scattering by pores and dense pre-sintered granules. In the low m% range, pores and granules are isolated, backscattering is negligible, and QM decreases with m% in proportion to scatterers number. As m% increases, reduction in QM is slowed because of backscattering. It is necessary to note that representation of QM as a function of concentration or porosity is not fully correct because of inherent frequency dependence of QM (i.e., different for each porosity or m%), and because QM characterizes a piezoelement (i.e., dimensions, frequency, capacity, etc.) instead of a material itself. A complete characterization also must include frequency dependencies of loss tan (i.e., the ratio of imaginary and real parts of corresponding elastic constants) for each concentration or porosity value. The full complex set of measured parameters for ceramic composites PZT/PZT are listed in Table 8. New ceramic piezocomposites composed by pre-sintered piezoceramic granules embedded in porous piezoceramic matrix were developed and investigated. PZT/PZT ceramic piezocomposites are characterized by a unique spectrum of the electrophysical properties

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unachievable for standard PZT ceramic compositions fabricated by standard methods and can be useful for wide-band ultrasonic transducer applications.

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Figure 24. Dependencies of mechanical quality factors of radial QMr and thickness QMt extensional modes of ceramic composite disks on m%.

Table 8. Complex set of parameters measured for thickness and radial extension modes of PZT/PZT composite disks for different concentrations of pre-sintered ceramic granules m% Parameter / mass. % Porosity, vol. % d33 quasi static (pC/N) fs rad (kHz) S’11E, 10-11 (m²/N) S’’11E, 10-14 (m²/N) S’12E, 10-12 (m²/N) S’’12E, 10-14 (m²/N) d’31 (pC/N) d’’31 (pC/N) ε'33T, 10-9 (F/m) ε''33T 10-12 (F/m) k'p k'’p e'31 (C/m²)

0 2.5 290 113.4 1.17  1.25  3.49 0.485 120  0.0974 11.20  3.93 0.561  0.0002 14.7

20 5.75 280 109 1.33  2.33  4.09 9.61 98  1.4713 9.35  56.32 0.473  0.0056 10.8

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60 20 280 84.4 2.60  19.44  7.77 32.06 45  1.7495 5.86  32.52 0.201  0.0015 2.6

100 30 260 70.3 4.48  56.73  13.22 43.21 20  2.0596 4.05  23.81 0.109  0.0073 0.8

154

A.N. Rybyanets Table 8. Continued

e''31 (C/m²) fs thick (kHz) k't k'’t C’33D, 1010 (N/m²) C’’33D , 108 (N/m²) C’33E , 1010 (N/m²) C’’33E, 108 (N/m²) e'33 (C/m²) e''33 (C/m²) ε'33S, 10-9 (F/m) ε'’33S, 10-9 (F/m)

 0.0038 2010 0.498  0.00899 17.2 7.71 13.01 21.2 19.6  0.1 7.62  0.81

 0.1480 1789 0.498  0.01842 12.48 12.24 10.19 21.9208 14.2  0.15 6.59  1.75

 0.0088 1076 0.516  0.02829 4.04 15.99 2.99 23.552 6.6  0.30 4.41  0.23

 0.0430 941 0.440  0.03118 2.33 19.89 1.91 25.3558 3.5  0.50 3.22  0.43

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8. OPTIMIZATION OF FINITE ELEMENT MODELS FOR POROUS CERAMIC PIEZOELEMENTS BY PIEZOELECTRIC RESONANCE ANALYSIS METHOD A novel approach for optimization of finite element modeling (FEM) of lossy piezoelectric elements was proposed in [72]. Procedure of optimization has consisted in sequential and iterative application of FEM and piezoelectric resonance analysis to complex electric impedance spectra of piezoceramic elements. For validation of proposed optimization procedures, FEM calculations of standard shape piezoelements (disks, shear plates, bars, and rods) made from porous PZT-type piezoceramics were fulfilled using FEM ANSYS software package. In recent years, low-Q piezoceramics and piezocomposite materials have been widely used for wide-band medical and NDT ultrasonic transducers with high sensitivity and resolution [1-6]. The majority of these advanced materials are lossy, and direct use of IEEE Standards for material constant determination leads to significant errors. The modeling and design of piezoelectric devices by finite element methods, among others, relies on the accuracy of the dielectric, piezoelectric and elastic coefficients of the active material used, which is commonly an anisotropic ferroelectric polycrystal. An accurate description of piezoceramics must include an evaluation of dielectric, piezoelectric and mechanical losses accounting for the out-of-phase material response to the input signal [58]. Standard finite element modeling (FEM) packages that are widely used for modeling of piezoelements and devices do not take into account these losses (although mechanical losses may be implicitly included in FEM calculations). Sets of material constants used for FEM calculations also do not include data on losses, except for QM for radial mode of vibrations. As a result, FEM calculations of real piezoelements and devices may give inadequate results for lossy materials (i.e., composites, porous ceramics, etc.) [73].

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8.1. Method of Measurements Numerous techniques using complex material constants have been proposed to take into account losses in low-QM materials and to overcome limitations in use of IEEE Standards for material constant determination [12, 18-20]. Iterative methods [19-20] provide a way to accurately determine complex coefficients in a linear range of poled piezoceramics from complex impedance resonance measurements. For our experiments, we applied the PRAP automatic iterative method [21, 68] to the full set of standard geometries and resonance modes needed to complete a characterization of hard PZT porous piezoceramics with moderate loss factors. This software uses a generalized form of Smits‘s method [19] to determine material properties for any common resonance mode, and a generalized ratio method for the radial mode [18] valid for all material Q‘s. The software routinely generates an impedance spectrum from the determined properties to indicate the validity of the results. Measurements of electric parameters were made on the Solartron Impedance/Gain-Phase Analyzer SL 1260.

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8.2. FEM Calculations and Optimization Procedure Finite element modeling of the effective constants of porous piezoelectric ceramics based on the effective moduli method was performed using ANSYS software package [73]. Different models of representative volume were considered: piezoelectric cubes with one cubic and one spherical pore inside, cubic volume evenly divided in partial cubic volumes, with a portion randomly declared as pores, and the like. For modeling porous piezoceramics with 3-0/3-3 connectivity type we used the representative volume with rigid skeleton structure (Figure 25) [56]. Optimization procedures were as follows: 1) Direct FEM calculations of complex electric impedance spectra using sets of material constants for standard piezoceramic elements. 2) Analysis of calculated impedance spectra by piezoelectric resonance analysis program (PRAP) and by deriving complex sets of material constants related to specific piezoceramic element. 3) Correction of initial material constants, including losses (Q = C’/ C’’, where C’ and C’’ real and imaginary parts of related material constants) Validation of optimization procedures was performed using the FEM ANSYS package. FEM-generated complex impedance spectra for standard-shaped piezoelements (disks, bars, shear plates, and rods) made from ― hard‖ porous PZT piezoceramics were processed by PRAP software to derive a complex set of material constants.

8.3. Results and Discussion Figures 26-29 show measured complex impedance spectra and PRAP approximations for the thickness (TE), radial (RE), length extensional (LE), and thickness shear (ST) modes of ― hard‖ porous piezoceramics with relative porosity 21% and average pore size 50 µm.

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a)

b) Figure 25. Examples of representative volumes used for FEM calculations of porous piezoceramics: (a) 3-3 connectivity, porosity 80%; (b) 3-0 connectivity, porosity 20%.

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Figure 26. Measured impedance spectra and PRAP approximations for TE mode of porous piezoceramic disk Ø20  1 mm2.

Figure 27. Measured impedance spectra and PRAP approximations for RE mode of porous piezoceramic disk Ø20  1 mm2.

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Figure 28. Measured impedance spectra and PRAP approximations for LE mode of porous piezoceramic rod 1.5  1.5  6 mm3.

Figure 29. Measured impedance spectra and PRAP approximations for ST mode of porous piezoceramic square 0.6  0.6  6 mm3.

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8.4. Full Set of Complex Material Constants Table 9 summarizes the complex constants of the porous ceramics obtained using PRAP analysis for a full set of standard geometries and resonance modes (Figures 26-29) . Additional physical parameters of porous ceramics measured by ultrasonic and hydrostatic weighting methods are as follows: ZA = 18.5 MRayl, ρ = 6.25 g/cm3, Vt = 2960 m/c., QMt = 150, (TC =340°) [6, 7].

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Table 9. Complex constants measured for various resonance modes of “hard” porous ceramics Parameter Real Imaginary Shear Thickness (ST) Mode fp (Hz) 1.60 ∙ 106  k15 0.56  0.013 CE55 (N/m²) 1.60 ∙ 1010 5.80 ∙ 108 CD55 (N/m²) 2.32 ∙ 1010 3.50 ∙ 108 D -11 S 55 (m²/N) 4.31 ∙ 10  6.49 ∙ 10-13 E -11 S 55 (m²/N) 6.25 ∙ 10  2.27 ∙ 10-12 e15 (C/m²) 7.51  0.14 h15 (V/m) 9.65 ∙ 108  1.32 ∙ 107 -10 d15 (C/N) 4.69 ∙ 10  2.56 ∙ 10-11 g15 (Vm/N) 0.04  0.001 εT11 (F/m) 1.13 ∙ 10-8  2.92 ∙ 10-10 εS11 (F/m) 7.77 ∙ 10-9  3.60 ∙ 10-11 Thickness Extensional (TE) Mode fp (Hz) 1.48 ∙ 106 kt 0.52  0.04 D C 33 (N/m²) 5.49 ∙ 1010 3.70 ∙ 108 E 10 C 33 (N/m²) 4.02 ∙ 10 2.66 ∙ 108 e33 (C/m²) 10.08  1.07 h33 (V/m) 1.47 ∙ 109  7.10 ∙ 107 εS33 (F/m) 6.87 ∙ 10-9  4.00 ∙ 10-10

Parameter Real Length Extensional (LE) Mode fp (Hz) 2.30 ∙ 105 k33 0.55 SD33 (m²/N) 2.12 ∙ 10-11 SE33 (m²/N) 3.03 ∙ 10-11 d33 (C/N) 2.54∙ 10-10 g33 (Vm/N) 0.04 εT33 (F/m) 6.85 ∙ 10-9 Radial Extensional (RE) Mode fp 1(Hz) 1.03 ∙ 105 SE11 (m²/N) 1.96 ∙ 10-11 E S 12 (m²/N)  6.37 ∙ 10-12 d31 (C/N) 5.92 ∙ 10-11 T ε 33 (F/m) 6.44 ∙ 10-9 kp 0.29 ζp 0.33 E S 66 (m²/N) 5.19 ∙ 10-11 E C 66 (N/m²) 1.93 ∙ 1010 Physical Parameters ρ (kg/m3) 6.25 ∙ 103 ZA (MRayl) 18.5

Imaginary   8.8 ∙ 10-4  5.92 ∙ 10-14  1.26 ∙ 10-14  1.35 ∙ 10-12  7.1 ∙ 10-5  1.5 ∙ 10-11   3.10 ∙ 10-14 2.30 ∙ 10-14  5.19 ∙ 10-14  3.09 ∙ 10-12  8.84 ∙ 10-5  6.3 ∙ 10-4  1.08 ∙ 10-13 4.0 ∙ 107  

8.5. FEM Results Figures 30-33 show complex impedance spectra generated by FEM using the constants shown in Table 9 and PRAP approximations for TE, RE, LE, and ST modes of porous piezoceramics.

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Figure 30. FEM calculated impedance spectra and PRAP approximations for TE mode of porous piezoceramic disk Ø20  1 mm2.

Figure 31. FEM calculated impedance spectra and PRAP approximations for RE mode of porous piezoceramic disk Ø20  1 mm2.

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Figure 32. FEM calculated impedance spectra and PRAP approximations for LE mode of porous piezoceramic rod 1  1  5 mm3.

Figure 33. FEM calculated impedance spectra and PRAP approximations for ST mode of porous piezoceramic square 0.6  0.6  6 mm3.

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The complex constants of ― hard‖ porous piezoceramics, obtained using FEM generated spectra (Figures 30-33), are summered in Table 10. It is readily seen from Tables 9 and 10 that the complex material constants resulting from FEM-generated impedance spectra are very close to measured ones, including imaginary parts (losses). For other low-Q materials, additional corrections of material constants (Q effective in ANSYS) with recurring optimization procedure should be performed. Table 10. Complex constants obtained using PRAP analysis of FEM-generated spectra for hard porous piezoceramics

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Parameter Real Shear Thickness (ST) Mode fp (Hz) 1.66 ∙ 106 k15 0.57 CE55 (N/m²) 1.67 ∙ 1010 CD55 (N/m²) 2.47 ∙ 1010 D S 55 (m²/N) 4.03 ∙ 10-11 SE55 (m²/N) 5.93 ∙ 10-11 e15 (C/m²) 9.49 h15 (V/m) 8.5 ∙ 108 d15 (C/N) 5.56 ∙ 10-10 g15 (Vm/N) 0.035 εT11 (F/m) 1.63 ∙ 10-8

Imaginary   0.048 1.48 ∙ 109 1.77 ∙ 108  2.89 ∙ 10-13  5.25 ∙ 10-12  1.26  2.4 ∙ 107  1.25 ∙ 10-11  0.0012  3.063 ∙ 10-9

εS11 (F/m) 1.12 ∙ 10-8  1.17 ∙ 10-9 Thickness Extensional (TE) Mode fp (Hz) 1.48 ∙ 106  kt 0.52  0.006 D 10 C 33 (N/m²) 5.49 ∙ 10 3.145 ∙ 108 E 10 C 33 (N/m²) 4.03 ∙ 10 5.92 ∙ 108 e33 (C/m²) 9.26  0.35 h33 (V/m) 1.57 ∙ 109  2.9 ∙ 107 S -9 ε 33 (F/m) 6.87 ∙ 10  3.29 ∙ 10-10

Parameter Real Imaginary Length Extensional (LE) Mode fp (Hz) 2.8 ∙ 105  k33 0.57  7.02 ∙ 10-4 D -11 S 33 (m²/N) 2.04 ∙ 10 1.038 ∙ 10-13 SE33 (m²/N) 3.04 ∙ 10-11  1.9 ∙ 10-13 d33 (C/N) 2.55∙ 10-10  1.27 ∙ 10-12 g33 (Vm/N) 0.04  1.48 ∙ 10-4 T -9 ε 33 (F/m) 6.48 ∙ 10  7.9 ∙ 10-12 Radial Extensional (RE) Mode fp 1(Hz) 1.03 ∙ 105  SE11 (m²/N) 1.99 ∙ 10-11  8.83 ∙ 10-14 SE12 (m²/N) -6.62 ∙ 102.53 ∙ 10-14 12

d31 (C/N) 6.37 ∙ 10-11 T ε 33 (F/m) 6.64 ∙ 10-9 kp 0.3 ζp 0.33 E S 66 (m²/N) 5.29 ∙ 10-11 E C 66 (N/m²) 1.89 ∙ 1010 Physical Parameters ρ (kg/m3) 6.25 ∙ 103 ZA (MRayl) 18.5

 8.42 ∙ 10-13  2.19 ∙ 10-11  0.002 0.0028  1.26 ∙ 10-13 4.49 ∙ 107  

9. SIMULATION OF ULTRASONIC WAVE PROPAGATION IN NONHOMOGENEOUS ANISOTROPIC CERAMIC COMPOSITES A comprehensive study including computer 3D simulation, impedance spectroscopy and pulse-echo ultrasonic measurements of various composite structures was carried out in [74]. The Wave 3000 Pro finite differences software package was used for simulation [75]. The results of the simulation of longitudinal ultrasonic wave propagation in different visco-elastic composites (porous ceramics, ceramics/ceramics and ceramics/crystals) were presented.

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Anomalies in sound velocities and attenuation near corresponding elastic dispersion regions were discovered. The simulation results were compared with the experimental data obtained by ultrasonic pulse-echo and through-transmit methods. Theoretical modeling, NDT inspection, and ultrasonic measurements of ceramic composites [2-6, 28] are very complex procedures. Changes in chemical composition on phase interfaces, as well as microporosity appearance during co-firing of composite components could alter elastic and mechanical properties of the composites. Spatial dispersion and scattering can distort ultrasonic pulse characteristics and make ultrasonic measurements ambiguous. Thus, other methods were proposed for composite structures modeling and evaluation [31, 56, 62].

9.1. Experimental Scenario

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The following two types of ceramic composites with strong spatial dispersion and high losses [5, 11, 71] were chosen as model samples for simulation of ultrasonic wave propagation and comparison with piezoelectric resonance analysis (PRAP) and ultrasonic measurements (Figure 34): 1. Ceramic composites consisting of soft PZT matrix with randomly distributed αAl2O3 crystals with a mean particle diameter ~ 200 μm and volume fraction from 9 up to 26 vol.% . 2. Porous PZT piezoceramics with different porosity and pore sizes (porosity 18 %, pore size 30 μm and porosity 21 %, pore size 75 μm).

a) Figure 34 (Continued).

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b)

c)

d) Figure 34. Optical and SEM micrographs of porous PZT piezoceramics (a, b) and PZT/α-Al2O3 composites (c, d) samples.

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9.2. Methods of Measurements Complex electrical coefficients of piezocomposite elements were determined by impedance spectroscopy method using Piezoelectric Resonance Analysis (PRAP) software [21, 68]. Measurements were made on Solartron Impedance/Gain-Phase Analyzer SL 1260. Sound velocity and attenuation of longitudinal and shear waves in different directions for polarized and non-polarized samples were measured by pulse–echo and through-transmit ultrasonic methods in the frequency range 1 ÷ 10 MHz using digital oscilloscope LeCroy Wave Surfer 422, Olympus 5800, 5077 pulser/receivers using standard Olympus transducers [74].

9.3. Simulations Algorithm The Wave 3000 Pro software package [76] was used for simulation. The program, besides simulating the complete spatial and time-dependent acoustic solution, allows simulation of ultrasound measurements in a variety of source and receiver configurations. The 3D composite objects for simulation were generated both internally, using Wave3000 Pro "Geometry" routines, and externally using optical and SEM slice data for real composites elements. Wave 3000 Pro computes an approximate solution to the three-dimensional visco-elastic wave equation. The numerical solution is based on a finite difference algorithm [77]. The algorithm was adapted to include viscous losses. The specific acoustic equation simulated in a Wave3000 Pro simulation is as follows:

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

    2w   (   w)      2 w        2 t 3 t  t  t  

In the equation above, which applies in an isotropic elastic region, the standard designations of variables are used [76]. Figures 35-38 show some examples of composite and porous ceramic models and corresponding through-transmit pulse characteristics simulated by Wave 3000 Pro program.

9.4. Results and Discussion 9.4.1. Ultrasonic Measurements Figures 39-42 show echo-pulse characteristics for porous ceramics and composite elements (diam. 25 mm, thickness 8 mm) measured at transmitted pulse frequencies 2.25 MHz and 5 MHz.

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Figure 35. Wave 3000 Pro models of PZT/α-Al2O3 composite.

Figure 36. Through-transmit pulse characteristics for PZT/α-Al2O3 composite simulated by Wave 3000 Pro.

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Figure 37. Wave 3000 Pro models of porous ceramics.

Figure 38. Through-transmit pulse characteristics for porous ceramics simulated by Wave 3000 Pro.

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Figure 39. Echo-pulse characteristics for porous ceramics element: transmitted pulse frequency 2.25 MHz; received pulse frequency 2.1 MHz.

Figure 40. Echo-pulse characteristic for porous ceramics element: transmitted pulse frequency 5 MHz; received pulse frequency 2.6 MHz.

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Figure 41. Echo-pulse characteristic for PZT/Al2O3 composite element: transmitted pulse frequency 2.25 MHz; received pulse frequency 0.9 MHz.

Figure 42. Echo-pulse characteristic for PZT/Al2O3 composite element: transmitted pulse frequency 5 MHz; received pulse frequency 0.93 MHz.

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It is readily seen from Figures 39-42 that because of ultrasonic wave scattering and spatial dispersion, the received echo-pulses are strongly distorted and shifted down in frequency. Thus, these pulse-echo measurements of elastic properties of ceramic composites as function of frequency are inaccurate and ambiguous.

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9.4.2. Resonance Measurements and Wave Pro Simulations Figure 43 shows frequency dependencies of real parts of elastic moduli С’33D and corresponding loss tanδ = С’’33D/С’33D for ceramic composites PZT/α-Al2O3 with different volume fractions of α-Al2O3 measured on standard disks samples by piezoelectric resonance (PRAP) method and simulated by Wave 3000 Prosoftware. It is readily seen that the elastic modulus С’33D increases with frequency because of frequency dispersion caused by the scattering of ultrasonic waves. It also can be seen that the frequency dependencies of the scattering loss tgδ are linear, which corresponds to stochastic scattering type tanδ ~ Df at λ ~ D. The decrease in the slope of С’33D versus frequency dependencies at α-Al2O3 volume fraction grows is caused by the decrease in the spatial inhomogeneity scale and the shift of the dispersion maxima to the high-frequency region. Figure 44 shows frequency dependencies of real and imaginary parts of elastic moduli C33D and C33E for porous piezoceramics measured on 1,3 and 5 harmonics of standard piezoceramic disks using PRAP analysis and simulated by Wave 3000 Pro program.

Figure 43. Measured and simulated frequency dependencies of elastic moduli С’33D and corresponding loss tanδ = С’’33D/С’33D for ceramic composites PZT/Al2O3 at different volume fractions of α-Al2O3.

It is readily observed that c’33E (real), as well as c’’33E (imaginary), increase with frequency because of the Rayleigh scattering (dispersion) of high frequency ultrasonic waves

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on pores. In turn, c’33D (real) decreases and c’’33D (imaginary) increases significantly with the frequency as result of electromechanical contributions to c33D (kt value measured on higher harmonics drops drastically). It is obvious from Figures 43 and 44 that Wave 3000 Pro simulations results are in good agreement with experimental frequency dependencies.

Figure 44. Measured on thickness mode harmonics and simulated frequency dependencies of real (C’33D, C’33E) and imaginary (C’’33D, C’’33E) parts of elastic moduli for porous piezoceramics disks.

CONCLUSION Over the past years, considerable advances were made to improve the physical, electrical and operational properties of piezoelectric ceramics using composite approaches. Numerous composite technologies were developed and a novel design idea was applied to develop functional piezoelectric ceramics. Commercialization of composite materials also lead to the development of new concepts of material and ultrasonic transducer design. This chapter presents a comprehensive review of microstructure peculiarities, mathematical models, methods of fabrication and measurements, as well as systematical experimental data for different types of ceramic piezocomposites. New families of polymer-free ceramic piezocomposites (porous ceramics, composites ceramics/ceramics and ceramics/crystals), are considered with properties that improve the combining of parameters of PZT, PN type ceramics and 1-3 composites. A line of proprietary porous piezoelectric ceramics was systematically studied. Complex sets of elastic, dielectric, and piezoelectric coefficients measured by piezoelectric resonance

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analysis methods were presented. It was shown that, for any connectivity type and porosity up to 70 %, the real structures of porous piezoelectric ceramics are close to the matrix medium structure with continuous piezoelectric ceramic skeleton. A critical comparison of numerical FEM calculations was carried out to provide the results of various approximated formulas and experimental data for different porous piezoelectric ceramics. It was shown that FEM calculations based on effective moduli methods give the most adequate results and agree well with the experiment in a wide porosity range. New low-Q ceramic piezocomposites technology based on the original microstructural design concept (MSD) using ― damping by scattering‖ approach was described. Complex sets of elastic, dielectric, and piezoelectric parameters of the ceramic piezocomposites were systematically studied using impedance spectroscopy and ultrasonic method. A line of wideband NDT ultrasonic transducers with high sensitivity and resolution was manufactured and tested. The main advantages of the new PZT/α-Al2O3 piezocomposites are high acoustic efficiency, low crosstalk, and low mechanical Q, and especially high acoustic sensitivity combined with well-damped signals. Additional advantages of the developed piezocomposites include the possibility of executing controllable changes in the main properties within a wide range, compatibility with standard fabrication technologies and processing flexibility. New ceramics and ceramic piezocomposites composed by pre-sintered piezoceramic granules embedded in porous piezoceramic matrix are described. PZT/PZT ceramic piezocomposites are characterized by a unique spectrum of the electrophysical properties unachievable in standard PZT ceramic compositions,. by novel fabrication methods and potential for use in wide-band ultrasonic transducer applications. Piezoelectric resonance analysis methods for automatic, iterative evaluation of complex material parameters were presented, along with full sets of complex constants for various ceramic piezocomposites. Critical comparisons of the results FEM calculations of effective constants for the ceramic piezocomposites with the results of various approximated formulas, unit cell models and experimental data were carried out. We also considered microstructural and physical mechanisms of losses and dispersion in ceramic piezocomposites, as well as technological aspects of its large-scale manufacture and application in ultrasonic devices . The results of this comprehensive study were presented, including finite difference 3Dsimulation, impedance spectroscopy, and pulse-echo ultrasonic measurements for different ceramic composites with strong spatial dispersion and high losses. A novel approach for optimization of finite element modeling (FEM) of lossy piezoceramic elements was also considered. Complex sets of material constants for ― hard‖ porous piezoceramics were obtained by PRAP analysis of electric impedance spectra measured on standard porous piezoceramic elements.

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[31] Hirschfeld, D.A.; Li, T.K., Liu, D.M. In: Key Engineering Materials; Dean-Mo Liu.; Trans. Tech. Publications: Switzerland, Vol. 115, 1996,65-80. [32] Rybyanets, A. In Advances in Applied Ceramics: Structural, Functional and Bioceramic; Edirisinghe M.; Maney Publishing: London, 2009 (will be published). [33] Newnham, R.Е.; Skinner, D.Р.; Klicker, K.A.; Bhalia, A.S.; Hardiman, B.; .R. Gururaja, T.R. Ferroelectrics, 1980, 27, 49-55. [34] Shгоut, Т.R.; Shulze, W.A.; Вiggегs J.V. Mat. Res. Bull. 1979, 14, 1553-1559. [35] Rybianets, A.N.; Tasker, R. Ferroelectrics. 2007, 360, 90-95. [36] Brown, W. F.; Trans. Soc. Rheol. 1965, 9, 357 –363. [37] Wiener, O. Abh. Sacks. Ges. (Akad.) Wiss. 1912, 32, 509. [38] Bruggeman, D. A. G. Ann. Phys. Lpz, 1935, 24, 636. [39] Wagner, K.W. Arch. Elektrotechn. 1914, 2, 371. [40] Marutake, M.; Ikeda, T.; J. Phvs. Soc. Jap., 1956, 11, 814. [41] Aleshin, V.I.; Tsihotsky, E.S.; Yatsenko, V.K. J. Tech. Phys. 2004, 74, 62. [42] Nan Ce-We. J Appl. Phys. 1994, 76, 1155-1163. [43] Rittenmyer, K.; Shrout, T.; Schulze, W.A.; Newnham, R.E. Ferroelectrics. 1982, 41, 189-195. [44] Banno H. Ferroelectrics. 1983, 50, 3-12. [45] Vorontsov, A.; Glushanin, S.; Topolov, V.; Panich, A. Abstracts of IV International Scientific conference «AIMS for future of engineering science». Montenegro, 2003, 40. [46] Rice, R.W. In: Key Engineering Materials; Dean-Mo Liu.; Trans. Tech. Publications: Switzerland, Vol. 115, 1996, 1-19. [47] Rice, R.W. Treatise on Materials Science and Technology. New York. 1977, 1, 191381. [48] Ramakrishman, N.; Arunachalam, V.S. J. Mat. Sci. 1990, 25, 3930-3937. [49] Liu, H.; Zhang, L.; Seaton, A. J.Colloid Interface Sci. 1993, 156, 285. [50] Wall, G.C.; Brown, J.C. J. Colloid Interface Sci. 1981, 82, 141. [51] Liu H., Seaton A. Chem. Eng. Sci. 1994. V. 49. P. 1869. [52] Avnir, D.; Farin, D.; Pfeifer, P. J. Chem. Phys. 1983, 79, 3565. [53] Tatarenko, L.N.; Tsikhotsky, E.S.; Yatsenko, V.K. Izv. Vusov. Sev. Kavk. Region. Estestv. Nauki. 2003, 40-44. [54] Hikita, K. H.; Jamada, K.; Nishioka, M.; Ono, M. Ferroelectrics. 1983, 49, 265-272. [55] Getman, I.; Lopatin, S. Ferroelectrics. 1996, 186, 301-304. [56] Nasedkin, A.; Rybjanets, A.; Kushkuley, L.; Eshel, Y. IEEE Ultrason. Symp. Proc. 2005, 1648-1651. [57] Rybianets, A.; Mogilevski, M.; Nudelman, I.; Kushkuley L. ICUltrasonics Proceedings. 2007, Paper 1298, 1-4. [58] Rybianets, A.N. Ferroelectrics. 2007, 360, 84 – 89. [59] Rybianets, A.; Eshel, Y.; Kushkuley, L. IEEE Ultrason. Symp. Proc. 2006, 19111914. [60] Rybianets, A. ECNDT 2006 Proceedings. 2006, Paper 750, 1-9. [61] Rybianets A. ICUltrasonics Proceedings. 2007, Paper 1163, 1-4. [62] Kim, J.-Y.; Ih, J.-G.; Lee B.-H. J. Acoust. Soc. Am.1995, 97, 1380-1388. [63] Gomez, T.E.; Montero, F.; Levassort, F.; Lesthiecq, M.; James, A.; Ringgard, E.; Millar, C.; Hawkings, P. Ultrasonics, 1998, 36, 907-923

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Biwa, S.; Watanabe, Y.; Motogi, S.; Ohno, N. Ultrasonics. 2004, 43, 5-12. Latiff, R.H.; N.F. Fiore. J. Acoust. Soc. Am. 1975,.57, 1441-1447 Bouda, Badidi; Lebaili, S.; Benchaala, A. NDTandE International, 2003, 36, 1-5. Gomez Alvarez-Arenas, T.E.; Mulholland, A.J., Hayward, G., Gomatam, J. Ultrasonics, 2000, 38, 897-907. Rybianets, A.; Eshel, Y.; Tasker R. 9th European Conference on Non-Destructive Testing (ECNDT 2006) Proceedings. 2006, Paper 129, 1-9. Rybianets, A.; Kushkuley, L.; Tasker R. The 9th Pacific Acoustic Conference WESPAC IX 2006 Proceedings. 2006, Paper 147, 1-9. Rybianets, A. 9th European Conference on Non-Destructive Testing (ECNDT 2006) Proceedings. 2006, Paper 750, 1-9. Rybianets, A., Motsarenko, T.; Eidelman, A.; Eshel Y. ICUltrasonics Proceedings. 2007, Paper 1160, 1-4. Rybianets, A.; Mogilevski, M.; Nudelman, I.; Kushkuley L. ICUltrasonics Proceedings. 2007, Paper 1298, 1-4. Nasedkin, A.V.; Rybianets, A.N. Izvestia, Sev. Kav. Reg., Ser. Tech. Science, Spec. Issue. 2004, 91-95. Rybianets, A.; Nudelman, I.; Eshel, Y. ICUltrasonics Proceedings, Paper 1296, 1-4. Rybianets, A.; Kushkuley, L.; Eshel, Y.; Nasedkin, A. IEEE Ultrason. Symp. Proc. 2006, 1533-1536, Wave 3000 Pro (Software for 3D Ultrasound Simulation). CyberLogic Inc. www.cyberlogic.org. Delsanto, P.P.; Schechter, R.S.; Mignogna, R.B. Journal Wave Motion. 1997, 26, 329-339.

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[77]

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In: Piezoceramic Materials and Devices Editor: Ivan A. Parinov, pp. 177-218

ISBN 978-1-60876-459-4 © 2010 Nova Science Publishers, Inc.

Chapter 4

SOME FINITE ELEMENT METHODS AND ALGORITHMS FOR SOLVING ACOUSTOPIEZOELECTRIC PROBLEMS A. V. Nasedkin1

Department of Mathematic Modelling, Southern Federal University2, Rostov-on-Don, Russia

ABSTRACT

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This chapter considers mathematical modeling by FEM of the piezoelectric devices. The formulation of dynamic problems in compound domains with different physical properties (piezoelectric, elastic and acoustic) is given. A new generalized Kelvin model for damping inputs in piezoelectric analysis is proposed. This model is completely compatible with the mode superposition method. When semi-discrete FEM approximations of the solution are applied to the governing equations in weak form, variational FEM equations with symmetrical saddle matrices are derived. A set of algorithms using symmetrical saddle matrices to create and solve FEM equations is proposed for static and dynamic problems. The Newmark method without velocities and accelerations node values is used for step-by-step time integration scheme; and modified Chollessky decomposition method is used as linear system solver. All procedures needed in FEM manipulations (the degree of freedom rotations, mechanical and electric boundary condition settings, etc.) also are provided in symmetrical form. FEM for evaluation of natural frequencies of compound electroelastic bodies are investigated. The schemes presented use FEM block matrices, where different matrix blocks are related to different field variables. The computer program ACELAN was developed based on these algorithms. The program was tested carefully and the results were compared with analogous values obtained by ANSYS, another well-known computer program. The numerical experiments showed that ACELAN and its algorithms are effective and give accurate results.

1 2

[email protected]. 8a, Milchakova Street, 344090 Rostov-on-Don, Russia.

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178

A. V. Nasedkin Finally, new mathematical models and numerical methods for modelling the dynamic behaviour of three-dimensional piezoelectric devices with rotation effects are presented. For piezoelectric vibratory gyroscopes working on ― energy trapped‖ effects, we introduce a small parameter, which is the ratio between rotation frequency and the principal resonance frequency. For modelling the work of piezoelectric gyroscopes, we use a series expansion of this small parameter. In the first phase, for zero approximation, we solve the eigenvalue and harmonic problems close to resonance frequency. The mechanical displacement results are stored in the nodes of finite element mesh for use in the next step. In the second phase, we solve the problem with axial rotation and relative displacement for resonance frequency, where Coriolis forces are considered as body forces. We also obtain the formulae for Coriolis forces concentrated in the nodes of finite elements.

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1. INTRODUCTION Dynamic problems of electroelasticity (piezoelectricity) for non-canonical areas in general cannot be solved by pure analytical methods and require application of numerical methods. Moreover, even for canonical areas, possible analytical methods often turn out to be so complicated and intricate that the application of numerical methods becomes justified. The main classes of such methods for solving boundary and initial-boundary problems for heterogeneous composite areas are the finite difference methods, which also include the variation-difference method, the finite element method (FEM) and, to lesser extent, the boundary element method (BEM). These methods are the primary methods for solving dynamical problems of electroelasticity. The first of the straightforward numerical methods applied for solving dynamic problems of electroelasticity seems to be the finite difference method [1]. Chulga and Bolkisel [2] apply the variation-difference method for solving the problems of electroelasticity in the case of steady oscillations and give the results of their computations. At present the most popular numerical method for solving the problems of electroelasticity is FEM. The derivation of FEM from the energy variational principal is first made in [3]. In numerous subsequent publications, FEM was given further serious development. Different kinds of FE were used and the technique for taking into account the boundary conditions for electrode surfaces was developed; special FE programs such as ACELAN [4], ATILA [5], CAPA [6], PZFlex [7, 8] were devised. These programs allow determination of all necessary characteristics of the fields, resonance and antiresonance frequencies, CEMC and the like. The capabilities of coupled piezoelectric analysis are also introduced in the range of universal FE packages such as ANSYS [9], COSMOS/M [10] and others. An overview of research made with the application of FEM before 1980 through 1990 is given in [11] and [2]. More recently, a very large number of publications devoted to research and application of FEM for piezodevices computation have appeared, making it impossible to give the full list here. We note only some early publications such as H. Allik, K.M. Webman, J.T. Hunt [12], D. Boucher, M. Lagier, C. Maerfeld [13], P. Challande [14], D.F. Ostergaard , T.P. Pawlak [15], D.R. Cowdrey, J.R. Willis [16], Y. Kagawa, T. Yamabuchi [17-18], R. Lerch [19], M. Naillon, R.H. Coursant, F. Besnier [20], and also the papers of J.-N. Decarpigny, R. Bossut,

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 179 P. Tierce, B. Hamonic and other authors, who are the members of ATILA developers team (see the list of 54 papers in [5]), PZFlex developers team [8, 21] and Kaltenbacher M. [22]. As a result, ready-to-use programs are now available for application in electroelasticity FEM research. Also, there is considerable experience in the practice of FEM computations of various piezoelectric devices. However, FEM development for electroelasticity problems of has been mainly in the direction of the extension of ordinary FEM approaches adopted for structural analysis. As a result, the number of necessary improvements in algorithmic realization of different blocks of FEM was kept to minimum. Therefore, these techniques are of limited use in piezoelectric finite element analysis. Even so, analogous software products are being developed. In 1997 at Rostov State University it was decided to develop ACELAN, a finite element complex oriented to piezoelectric devices computations. Several stages of ACELAN development are described in [23-27] and other publications. A set of original algorithms was developed especially for ACELAN, for use in solving matrix problems of FEM for FE approximations in problems involving electroelasticity, thermoelectroelasticity and acoustic electroelasticity. These algorithms take into account the specificity of FE equations arising from computations of piezoelectric devices. This chapter describes the main methods of piezoelectric materials modeling adopted in ACELAN.

2. ACOUSTOPIEZOELECTRIC PROBLEM STATEMENTS AND FINITE ELEMENT APPROXIMATIONS

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Let us assume that the piezoelectric device consists of a piezoelectric medium    p ;

   is the boundary of the region  ; n is the vector of the external unit normal to  . We also assume that the region  and its boundary  are subjected to the following conditions:  is the sum of a finite number of sets, star-shaped with respect to any spheres 1 contained in them; while  is a Lipschitz boundary of class C . These (  ,  ) usual mathematical conditions for elastic problems are detailed in [28].

2.1. Classical and Weak Formulations for Piezoelectric Problems We also assume that the physical-mechanical processes, which take place in piezoelectric media  , are determined by the displacement vector function u ( x , t ) and the function of electric potential  ( x , t ) . With knowledge of these functional fields, one can determine the tensor of mechanical stresses ε (u ) and the vector of electric field E( ) , as follows:

ε  (u  u* ) / 2 , where 

E  

(1)

is the nabla-operator, and in R3 in the Cartesian coordinate system

  { / x1,  / x2 ,  / x3} ; * is the conjugation operation.

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180

A. V. Nasedkin

The governing relations for piezoelectric material connect the tensor of mechanical stresses σ and the vector of dielectric displacement D with ε and E

σ  cE  ε  d ε   e*  E ,

(2)

  e   (ε   ε )  ε S  E , D  d D d

(3)

E where c is the fourth rank tensor of elastic moduli, measured for a constant electric field; e S is the third rank tensor of piezoelectric moduli; ε is the second rank tensor of permittivity

moduli for a constant mechanical strain;

d ,  d are the damping coefficients; (...)   (...)

E is the double convolution ( c E  ε  cklmn  mn ek em ; ek is the basic vector of the Cartesian coordinate system). The equation of motion and equation of quasielectrostatics can be written in the form

  d u ) ,   σ   f   (u

(4)

 D   ,

(5)

where  is the mass density of the material; f is the vector of mass forces;  d

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damping coefficient;   is the density of free charges (usually,

is the

   0 ).

In models (1)–(5) for the piezoelectric material, we use a generalized Rayleigh method of damping evaluation [25, 26], which is admissible for many practical applications. When  d  0 in Equation (3), we have the usual model for taking into account of mechanical attenuation in piezoelectric media which is adopted in several well-known finite element packages such as ANSYS, COSMOS/M, and others. It is true that, by virtue of the coupled state of the mechanical and electric fields, the damping effects will also extend into the electric fields when  d  0 . The more complicated models (2), (3) extend the Kelvin's model to the case of piezoelectric media. It has been shown that the model (2), (3) with  d   d satisfies conditions which ensure the dissipation of energy and the possible splitting of the finite element system into independent equations for the modes [26]. We suppose that the density function  ( x ) is piecewise-continuous and   0  0 :

 ( x )   0 . The other moduli in (2), (3) are piecewise uninterrupted with the first derivatives from x with usual symmetry properties: E E E , cijkl  cEjikl  cijlk  cklij

eikl  eilk ,

 klS  lkS .

(6)

In addition, for positive definiteness of intrinsic energy for piezoelectric medium, the following inequalities must be satisfied

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 181

W0  0 , ε  ε* , E :

ε cE ε  E  εS  E  W0 (ε ε  E  E) .

(7)

The system of differential equations (4), (5) should be completed by boundary and (in the transient case) initial conditions. The boundary conditions are of two types: mechanical and electric. In order to formulate mechanical boundary conditions, let us assume that the boundary    is divided to the subdomains  and u (     u ). On the part  , there is given the vector of mechanical stress p 

p  n  σ , p  p ,

x   .

(8)

Suppose that u  uj ; j  1, 2,..., L ;  u 0   , uj do not border one another; while among uj there are L  1  l surfaces with determined functions of displacement u  u* ( j  J r  {0, l  1, l  2,..., L} ) and l plane regions ( j  J p  {1,2,..., l} ), in contact with rigid massive punches (stampes). We will connect with region uj , j  J p the local coordinate system O( j )1( j )2( j )3( j ) so that the axis

3( j ) coincides in direction with the

direction of external normal n at the point O( j ) ; and the axes

1( j ) and 2( j ) will be the

main axes of inertia for the punch with number j . Then, we can assume the following boundary conditions for uj 2

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n  u   ujkk( j ) , ( 0( j )  1 ), k 0



uj

x uj ,

 p( j )n  σ  n d    ujpm(p j )  Pjp ,

n  σ  (n  σ  n) n  0 ,

u  u* ,

x uj ,

jJp p  0,1, 2 ,

x uj , u 0   ,

(9)

jJp

jJp j  Jr ,

where in (9), (10) the summation by repeating index j and p is missing; displacement of the punch with number j ; angles about axes

(10) (11) (12)

 uj0 is the normal

 uj1  2( j ) ,  uj 2  1( j ) are the punch rotation

2( j ) and 1( j ) , respectively; m(0j ) is the mass of punch; m(1 j )  J 

( j) ( j) 2 2

,

m(2j )  J  ( j ) ( j ) are the inertia moments of punch; Pjp  Pjp (t ) are the force (with p  0 ) 1

1

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182

A. V. Nasedkin

and the moments (with p  1, 2 ), acting on the punch with number j ; u*  u* (x, t ) are the components of determined functions of displacements. Let us pass to a formulation of electric boundary conditions. We assume that the boundary  is also subdivided to the parts D and  (   D   ). The parts D are assumed to be free of electrodes, and the surface density of electric charge   is assumed to be known:

n  D    ,

x  D ,

(13)

and usually,    0 . The

subset



itself

is

subdivided

into

M 1

( j  J eo  J es , J eo  {1,2,..., m} , J es  {0, m  1,..., M } ,  j 0   ,

subdomains which

are

 j not

adjacent to one another and are coated by infinitely thin electrodes. We specify the following boundary conditions on these areas as follows:

   j (t ) ,



 j

x  j ,

n  D d   Q j (t ) ,

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  V j (t ) , x  j ,

j  J eo ,

(14)

Qj  I j ,

j  J eo

(15)

 0   ,

j  J es .

(16)

where  j , Q j , V j depend only on time t ; and the sign ― ‖ in (15) is chosen in accordance with the accepted direction of the current in the electric circuit. By (14), (15) there are m electrodes on which the electric potentials  j are initially unknown, but the overall electric charges Q j or currents I j on each electrode are defined. The remaining M  1  m electrodes are the electrodes with known electrical voltages V j . Let us note that  j in (14) and V j in (16) are independent of space coordinate x , and therefore, the electrodes are equipotential surfaces. Integral condition (15) is an analogue of the contact condition (10) for rigid massless punches ( m(pj )  0 ). But the distinguishing feature of the piezoelectric elements is that boundary conditions (14) – (16) are necessary for them, since they determine the outer electric influence between electric potentials and charges or circuits. Note that for homogeneous boundary conditions on the electrodes  j with Q j  0 in (15) and V j  0 in (16), these surfaces are called open-circuited electrodes and shortcircuited electrodes, respectively.

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 183 We also assume that all domains  , uj ,  D ,  j have Lipschitz boundaries of class

C1 [28]. as

For transient problems it is also necessary to pose initial conditions, which can be written

u(x, 0)  u* (x) , u(x, 0)  u* (x) ,

x

(17)

where u* (x) is the initial displacement and u* (x) is the initial velocity of the body‘s points

x.

Formulas (1)–(17) represent a classical formulation of the linear problems of piezoelectricity with generalized Rayleigh damping. Note that in this setting, the boundary conditions (9)–(11) usually are absent. We introduce these unconventional cases of contact conditions with rigid punches to demonstrate an analogy between these mechanical boundary conditions and electric boundary conditions for electrode surfaces. To formulate the weak or generalized solution of dynamic problem for piezoelectric solids, we begin by introducing the space of vector functions u  {u1, u2 , u3} and function

 , defined in  , which we will need later.

2 We will denote by H0 the space of vector functions u  L with scalar product:

( v, u) H0   vi ui d  . 

(18)



1

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For the set of vector functions u  C which satisfy homogenous boundary conditions (12) for uj , j  J r , i.e.

x uj ,

u  0,

and (9) for arbitrary

u 0   ,

j  Jr ,

 ujk for uj , j  J p , we introduce the scalar product

( v, u)H1   vi ,k ui ,k d  . ul

(19)



(20)

The closure of this set of vector functions u in the norm generated by scalar product (20) will be denoted by H1ul , where l is the number of boundary conditions (9)–(11) for rigid punches. For the set of functions

  C1 which satisfy homogenous boundary conditions (16) for

 j , j  J es , i.e.

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A. V. Nasedkin

  0,

x  j ,

 0   ,

j  J es .

(21)

and (14) for arbitrary  j for  j , j  J eo , we introduce the scalar product

(  , )H1   ,i,i d  .

(22)



m

The closure of this set of functions  in the norm generated by scalar product (22) will be denoted by H1 m , where m is the number of boundary conditions (12). Finally we introduce the spaces Qul and Q m

Qul  L2 (0, T ;H1ul ) ,

Qm  L2 (0, T ;H1m ) ,

(23)

2 where for Banach space X with norm ||  ||X , the space L (0, T ; X ) is the space of (class)

functions t  f (t ) from [0, T ] into X , which satisfies the condition T

(  || f (t ) ||2X dt )1/ 2 || f ||L2 (0,T ; X )   0

We present the solution {u,  } of the transient problem (1)–(17) for piezoelectric medium in the form

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u  u0  u  ,

  0    ,

(24)

where u 0 ,  0 satisfy homogeneous boundary mechanical and electric conditions and u ,

 satisfy the inhomogeneous boundary conditions, i.e.: 2

2

k 0

k 0

n  u0   ujk0k( j ) , n  u   ujk k( j ) ,  ujk0   ujk   ujk , x uj , j  J p

(25)

u0  0 ,

(26)

u   u * ,

x uj ,

0  0 j (t ) ,   j (t ) , x  j , 0  0 ,

  Vj (t ) .

 0   ,

u 0   ,

j  Jr ,

0 j  j   j , j  J eo ,

(27)

x  j ,

(28)

j  J es .

and therefore

u 0  Q ul , 0  Qm .

  QM .

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(29)

Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 185 To formulate the weak or generalized solution of dynamic problem for piezoelectric solid we realize the following operations. At the beginning we scalar-multiply Equation (4) by some arbitrary vector function v  H1ul . Further we add together Equation (5) and differentiated by time and multiplied by  d Equation (5). After that we multiply received formula by some arbitrary function

  H1m . By integrating the obtained equalities over

domain  and using integration by parts, and taking into account the formulated boundary conditions, we obtain the weak formulation of dynamic problem for piezoelectric solid. Definition 2.1. The functions {u,  } in the form (24) with (25)–(29) u 0  Q ul ,

0  Qm ) are the weak solution of the dynamic problem of the piezoelectric media, if the following integral relations are satisfied:

 (v, u0 )  d (v, u0 )  c(v, u0 )  e(0, v)  Lu (v) ,

(30)

 d e(  , u0 )  e(  , u0 )   (  ,0 )   d L e (  )  Le (  )  L (  ) ,

(31)

  H1m , and the initial conditions (17) also hold.

for t  [0, T ] ; and v  H1ul ,

Here we introduce the bilinear forms and functions of function 2

l

 ( v , u )   ( v, u )  



j 1

k 0

v jk

 ujk mk( j ) ,

 ( v, u)  ( v, u)H , 0



E ij ( v) kl (u) d  , c( v, u)   cijkl

(33)

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

e( , v)   eikl kl ( v),i d  , 

 (  , )    ijS ,i, j d  , 

d (v, u)  d  (v, u)  d c(v, u) , l

Lu ( v)  L fp ( v)   j 1

2

 k 0

v jk

(34) (35)

Pjk   ( v, u  )  d ( v, u  )  c( v, u  )  e(  , v) ,(36)

L fp ( v)   v  f d    v  p d  , 

(32)

(37)



m

L e (  )     d      d    X jQ j  e(  , u ) ,

(38)

L (  )   (  , ) ,

(39)



D

j 1

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186

A. V. Nasedkin

where

 vjk are the values from (9) for vector-function v  H1ul ; and X j are the values from

(14) for function

  H1m .

From this weak formulation of the transient problem, we can easily obtained the weak formulation of the static problem, modal problem and harmonic problem for the piezoelectric solid. When all external influences — specified boundary pressures, forces, charges, displacements and potentials — vary along with the same harmonic law exp[ j t ] , we have the behavior of steady-state oscillations: u  u exp[ j t ] ,    exp[ j t ] . In this case, as is obvious from (30), (31), we have a system of integral relations for the amplitude functions and values

2 (v, u0 )  jd (v, u0 )  c(v, u0 )  e(0 , v)  Lu (v) ,

(40)

1  (  ,0 )  L 0 (  ) , (1  j d )

(41)

e(  , u0 ) 

where the symbol (  ) for amplitude functions and values is absent, u0  H1ul ,

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L 0 (  )  L e (  ) 

0  H1m ,

1 L  (  ) . (1  j d )

(42)

2.2. Modal Problems: Mathematical Properties and Some Theorems about Eigenfrequencies In this section we focus on eigenvalue problems for piezoelectric solids. Such problems are central to the analysis of real piezoelectric devices working in dynamic conditions. Indeed, the frequencies of electric resonances and antiresonances are the natural frequencies of a piezoelectric body. These frequencies determine the dynamic electromechanical coupling factors and the most effective frequency ranges for real piezoelectric device. The resonance frequencies f k  k /(2 ) for piezoelectric solids can be found from the solution of the generalized eigenvalue problem or modal problem, obtained from (40), (41) with d ( v , u 0 )  0 ,  d  0 , Lu (v)  0 , L 0 (  )  0 , i.e., without any external inhomogeneous influences and without damping effects (  d  0 ,  d  0 ,  d  0 )

2 (v, u)  c(v, u)  e(, v)  0 ,

(43)

 e(  , u )   (  ,  )  0 ,

(44)

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 187 where u  u0 ,    0 , since u   0 ,    0 for homogeneous boundary conditions. Problem (43), (44) is an eigenvalue problem and consists of finding the eigenvalues

   2 and eigenfunctions {u,  } , which give non-trivial solutions of the homogeneous boundary problem. We can transform the system (43), (44) (and also (40), (41) for harmonic problem) by eliminating the functions  . Note that by virtue of the properties assumed earlier, the forms

 ( v, u ) , c ( v, u ) and  (  ,  ) are symmetrical, bilinear and positive defined in L2 , H1ul , H1 m respectively, while e( , v ) is only a bilinear form. Since for fixed u  H1ul ,

  H1m , the forms e(  , u) and  (  ,  ) are linear-bounded

functions in H1 m , by Riesz's theorem, the elements eu , Thus, for

  H1m exist and are unique.

  H1m

e(  , u)  (  , eu) H1 ,

(45)

 (  , )  (  ,  )H .

(46)

m

1

m

It is obvious that eu and  are linear operators acting from H1ul into H1 m and from

H1 m into H1 m , respectively; and that an inverse exists for the operator  . From (44)–(46) we obtain that

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  eu ,   Alm u ,

Alm   1e ,

(47)

where the operator Alm acts from H1ul into H1 m , and is linear and bounded. Using (45)–(47), we can represent the system (43), (44) in the reduced form

2 (v, u)  clm (v, u)  0 ,

(48)

where

clm (v, u)  c(v, u)   ( Almv, Almu) .

(49)

Definition 2.2. We will call the set of quantities {  , u  H1ul , 2

  H1m } which

satisfies (48) for arbitrary vector function v  H1ul , or which is equivalent (43), (44) for arbitrary v  H1ul ,

  H1m , a generalized solution of eigenvalue problems for a

piezoelectric body.

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188

A. V. Nasedkin By repeating the topics presented in [28], we can show that the space H1c , which is the 1

closure of the set of vector function u  C , satisfying (9) and (19) in the norm generated by the scalar product (49), is equivalent to H1ul ; and the next two theorems follow from the complete continuity of the embedding operator from H1ul into H0 , as also in the general situation [29]. Here the form c( u, u)   (  ,  ) should be positive defined, which is provided by conditions (7). Theorem 2.1. The operator equation (48) has a discrete spectrum 0  12  22   

2k   ; k2   as k   ; and the corresponding eigenfunctions u( k ) form a system that is orthogonal and complete in the spaces H0 and H1c . Theorem 2.2. (The Courant–Fisher minimax principle).

 2k 

   , min ( v ) R lm v 0, vH1ul w1 , w 2 ,..., w k 1H1ul     ( v ,w j ) 0; j 1,2,...,k 1 max

where Rlm ( v ) is the Rayleigh quotient

Rlm ( v) 

clm ( v, v) .  ( v, v)

(50)

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Proofs of these two theorems completely repeat well-known proofs of corresponding theorems for conventional elastic media with the replacement of forms c and  by clm and

 [29]. We observe that the orthogonality conditions in Theorem 2.1 can be presented in the forms

(u(i ) , u( j ) )H0  0 , (u(i ) , u( j ) )H1  0 , 

i  j.

(51)

c

Then, we observe the change of natural or resonance frequencies of problem (43), (44) or (48), when some of their parameters change. These changes will be indicated explicitly in formulations of the following theorems, and all the quantities referring to the modified problems will be indicated by subscripts lm or by an asterisk. For initial and modified problems not specified in formulations of the theorems, the determining parameters are assumed to be identical. We will also call problem (43), (44) or (48), the lm-problem, emphasizing by this the presence of l areas uj , j  1, 2,..., l in contact with rigid plane punches and m opencircuited electrodes  j , j  1, 2,..., m .

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 189 We will consider two similar lm- and pm-problems, which differ solely in the number l and p of contacting surfaces of uj in (9)–(11). All the remaining input data from (1) – (16) in the lm- and pm-problems are assumed to be the same. Theorem 2.3. If 0  l  p  L , for any k , the kth natural frequency is no less than kth natural frequency

 pmk of pm-problem, i.e., 

2 lmk



lmk of lm-problem 2 pmk

.

Note that in conditions of Theorem 2.3, we do not change the boundary u . When passing from lm-problem to pm-problem, we change only the conditions of fixed boundary to the conditions of contact with punches on the part of uj . We now consider two similar lm- and ln-problems, which differ solely in the number m and n of open-circuited electrodes of  j in (14)–(16). Theorem 2.4. If 0  m  n  M , for any k, the kth natural frequency problem is no greater than the kth natural frequency

lmk of the lm-

l nk of the ln-problem, that is,

2lmk  2l nk . Theorem 2.5. If for the two problems the rigid fixed boundaries and the boundaries, contacting with the punches, such that  u  *u , uj  *uj , j  0,1, 2,..., L , then we have

2lmk  *2lmk for all k.

Theorem 2.6. If the elastic moduli, the piezomoduli, the densities and masses, and the inertia moments of the punches of the two problems are such that clm ( v, v)  c*lm ( v, v) ,

 (v, v)  *( v, v) for  v  H1ul , then 2lmk  *2lmk for all k.

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Theorem 2.7. If the electrode boundaries of two problems are such that   * ,

 j  * j , j  0,1, 2,..., M , then we have 2lmk  *2lmk for all k. Theorem 2.8. If the permittivities of two problems are such that

 (  ,  )  * (  ,  )

   H1m , then 2lmk  *2lmk for all k. The proofs of these theorems are presented in [30]. Now we will summarize the results of Theorems 2.3 – 2.8. If on certain areas of uj we replace the boundary conditions of rigid clamping (12) by the contact boundary conditions (9)–(11), then, by Theorem 2.3, the natural frequencies can only decrease. On the other hand, if on certain areas of  j we replace the boundary conditions for the electric potential of zero (16) with electric boundary conditions of contact type (14), (15) for open-circuited electrodes, then by Theorem 2.4 the natural frequencies can only increase. Note that the natural frequencies in the problem with all the short-circuited electrodes are usually named as electric resonance frequencies, while the natural frequencies in the problem with some open-circuited electrodes are usually named as electric antiresonance frequencies. Therefore, Theorem 2.4 also asserts that electric antiresonance frequencies are not less than the electric resonance frequencies with the same order numbers.

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190

A. V. Nasedkin By Theorems 2.5 and 2.6, a reduction in the boundaries uj or a specific reduction in the

elastic moduli and an increase in the density or in the massive characteristics of punches can lead only to a decrease in the natural frequencies. Conversely, by Theorems 2.7 and 2.8, a reduction in the electrode boundary  j or a specific reduction in the permittivity moduli can lead only to an increase in the natural frequencies. It can be noted that changes in mechanical conditions lead to known changes in natural frequencies [29], which have clear physical explanations. Thus, if the body is more rigidly fixed, then its mechanical vibrations become more constrained; therefore, eigenvalues can only increase. If the changes in moduli and physical properties are such that the potential energy increases and kinetic energy decreases, then the natural frequencies also increase. Comparing the effects reflected in Theorems 2.3, 2.5, 2.6 and 2.4, 2.7, 2.8, we can conclude that a similar change in the mechanical and electric boundary conditions or in elastic and permittivity moduli leads to an opposite change in the natural frequencies.

2.3. Finite Element Approximations for Piezoelectric Problems For solving transient problems (30), (31), (17); harmonic problems (40), (41) or modal problems (43), (44) in the weak forms, we will use the classical finite element approximation techniques [31, 32]. Let  h be a region of the corresponding finite element mesh h   ,

h 

k

ek , where  ek is the separate finite element with number k. On the boundary

 h    h , we can introduce the boundaries  h ,  hu , huj , and so on, which are

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approximations of the corresponding boundaries  , u , uj , etc. Then, for  h with the suitable boundaries, we can define the functional spaces H1hul , H1h m and Qhul , Qh m in a manner similar to that used for the functional spaces in section 2.1. On the finite element mesh h 

k

ek , we find the approximation to the weak

solution { u h  u 0 ,  h   0 } for transient problem (30), (31) in the form

uh (x, t )  N*u (x)  U(t) ,

h (x, t )  N* (x)  Φ(t ) ,

(52)

where N*u (x) is the matrix of the shape functions for displacements, N* (x) is the row vector of the shape functions for electric potential, and U ( t ) , Φ ( t ) are the global vectors of nodal displacements and electric potential, respectively. All shape functions form the basis of t the corresponding finite-dimensional spaces Vhul  Qhul and Vhtm  Qhm for transient

problems or, in the finite-dimensional spaces Vhul  H1hul and Vhm  H1hm , for harmonic t and modal problems. Then, for transient problems, uh  Vhul ,

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h  Vhtm .

Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 191 .

In (52), for harmonic and modal problems, U and Φ are constant vectors. We represent the projecting functions v and  in finite-dimensional spaces Vhul and

Vh m , respectively, by the formulae

v(x)  N*u (x)   U ,

 (x)  N* (x)   Φ .

(53)

In accordance with conventional finite element technique, we approximate the continual weak formulation (30), (31) by the problem in finite-dimensional spaces. Substituting (52), (53) into problem (30), (31) with (32)–(39) for  h , we obtain

Muu  U  Cuu  U  Kuu  U  Ku  Φ  Fu ,

(54)

 d K*u  U  K*u  U  K  Φ   d F e  F e  F ,

(55)

with the initial conditions

U(0)  U* , U(0)  U* ,

(56)

which are derived from the corresponding continual conditions (17). Here, Muu  Muu  M p , Muu 

M a k

ek uu

, K uu 

K a k

ek uu

, etc., are the global

matrices, obtained from the corresponding element matrices ensemble (



a

), and M p is

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the matrix of punch mass and inertia characteristics. According to (32)–(39), the element matrices are in the following form ek Muu 

ek

ek ek ek ek ,  Nue  Nue* d  , Kuu   Sue*  c E  Sue d  , Cuu  d Muu  d Kuu ek

(57) ek K uek   ek Sue*  e*  Se d  , K   ek Se*  ε S  Se d  ,

(58)

Sue  L()  Nue* ,

Sue   Ne* ,

(59)

1 0 L ()   0  2   0 0

0 0 3 2  0  3 0 1  ,   3  2 1 0 

(60)



*



where Nue* , Ne* are the matrix and the row vectors of approximate shape functions, respectively, defined on separate finite elements.

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192

A. V. Nasedkin E In (57)–(60), we use the vector-matrix forms for moduli: c is the 6x6 matrix

E E (pseudomatrix) of elastic moduli, c ,  ,   1, 2,..., 6 , i , j , k , l  1, 2, 3 with  cijkl

correspondence

law

4  (23)  (32) ,

  (ij ) ,

  ( kl ) , 1  (11) ,

5  (13)  (31) ,

6  (12)  (21) ;

2  (22) , e

is

the

3  (33) , 3x6

matrix

(pseudomatrix) of piezoelectric moduli ( ei  eikl ). *

Here, in vector–matrix notation, we apply the symbol a  b for scalar product of two vectors that is conformed to the multiplication operation of matrices by vectors and the general equations (1)–(5) for piezoelectric solid can be represented in the following form:

S  L()  u ,

E   ,

T  cE  (S  d S)  e*  E ,

(62)

D   d D  e  (S   d S)  εS  E ,

(63)

L* ()  T   f   u  d  u ,

(64)

*  D   ,

(65)

S  {S1, S2 , S3, S4 , S5, S6}  {11, 22 , 33,  23,  13,  12} , T  {T1, T2 , T3, T4 , T5, T6}  {11,22 ,33,23,13,12} , Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

(61)

 ij  2ij ,

(66) (67)

where S is the strain vector (pseudovector) and T is the stress vector (pseudovector). ek ek Note that here the global and local mass and stiffness matrices M uu , Muu , K uu , K uu ek are formed as in the case of pure elasticity, and the matrices K u , K uek , K , K  are the

usual finite element piezoelectric stress and dielectric permittivity matrices for piezoelectric body. The vectors Fu , F e , F e , F in (54), (55) are obtained from the right-hand sides of (36)–(39), (53) and finite element representations of u and

 . By virtue of the positive

definition of potential energy (7) the symmetric matrices K uu , K and also M uu are positive definite.

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 193

2.4. Acoustic Media with Dissipation Modelling We start with a system of equations describing the distribution of acoustic waves of small amplitude in a liquid medium with dissipation [33] as follows:

1 p   v  0 ,  wcw2

rot v  0 ,

(68)

wv    w ,

w   pI  bv,

(69)

where p is the overpressure or acoustic pressure; v is the vector of fluid velocity (oscillatory velocity);  w is the mean fluid density; cw is the speed of sound in fluid medium; σ w is the second rank stress tensor in fluid medium; I is the second rank unit tensor; b is the dissipative coefficient. The subscript ―w ‖ in (68), (69) denotes affiliation to the properties of fluid medium. Here we address liquid and gas media in the network of common acoustic approach. Let us assume that the fluid fills out the volume with the boundary w   w . We also

assume

w  wf

that

wc

the

wi

boundary

w

is

divided

to

the

four

subdomains:

ws . Let us also assume that  wf is the free boundary,  wc is the

rigid boundary, wi is the boundary with an impedance condition or damping boundary, and

ws is the part of the boundary with fluid-structure coupling. In this case, the following

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boundary conditions occur:

v0,

x   wc ,

(70)

nw  σw  0 ,

x wf ,

(71)

nw  σw  Z v ,

x   wi ,

(72)

n w  σ w  n w  σ s , v  u , x   ws ,

(73)

where n w is the unit normal to  w which is external with respect to w ; Z is the impedance of the boundary wi ; σ s and u are the stress tensor and displacement vector of the solid. From the second formula (68), we introduce the velocity potential 

v   and from (68), (69) we obtain the following equation:

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(74)

194

A. V. Nasedkin

1 b .      w  0 ,  w  2  wcw2 cw

(75)

Further, we express the acoustic pressure p and the fluid stress tensor σ w from potential as follows:

p  w  b  ,

(76)

σw  w I  b  I  b  .

(77)

Let us multiply Equation (75) by arbitrary, sufficiently smooth function

    x . By

integrating the obtained equalities over domain w , using integration by parts after standard integration by parts, and taking into account Equation (74), we obtain:

1  d       d    w     d     nw   v   w v  d  . (78) w w w cw2 w Unfortunately, the integral in the right-hand side of (78) is badly subsumed to satisfy the boundary conditions (71)–(73). We shall note some properties, supporting to avoid this difficulty. In the statement of considered acoustic problems, we assume a small absorption of distance sound comparable to the wave-length. For example, the coefficient  w from (75), for water with absorption

 / f 2  2 2 w / cw  2 1014 (s2/m), is equal to  w  0.15 1011

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(s), and then b  0.3  10

2

( kg/(m  s) ). By simple dimensional analysis, we determine that

in these problems, the parameter  w is small. Moreover, in fact the boundary conditions (71)–(73) are satisfied only approximately, particularly taking into account the terms with coefficient  w . Thus, without a large loss of accuracy in realization of the boundary conditions (71)–(73) for σ w from (77), it is possible to approximate [34] as follows:

σw  σaw  w    w  I .

(79)

By using (79), (74), we approximate boundary conditions (71)–(73) as follows:

 0,

x wf ,

(80)

nw   v   wv   Z 1w   2 w  ,

x   wi ,

(81)

nww    w   nwσs ,

x ws ,

(82)

v u,

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 195 where in (81), the term with

 w2 is omitted.

1 Then, on the set of functions   C defined on w which satisfy homogenous

boundary conditions (80) on  wf , we introduce the scalar product

(, )H1   ,i ,i d  . 

(83)

w

The closure of this set of functions  , in the norm generated by scalar product (83), is denoted by H1 . As in section 2.1, we introduce the space Q  L2 (0, T ;H1 ) . From Equations (70), (80)–(82) for integral in right-hand side of (78) multiplied by  w , we obtain the resulting integral identity for   Q as

w (, )  dw (, )  cw (, )   wr(, u)  r(, u)  0 ,

(84)

where

 w ( , ) 

w cw2



w

 d  

2  w2 w Z

d w (, )   w  w     d   Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

w

cw (, )   w     d  , w



 wi

 w2 Z

 d  ,

(85)



(86)

wi

 d  ,

r(, u)   w   nw  u d  ws

(87)

for   H1 . We now apply the usually semi-discrete finite element approximation in the finite element space, defined on h w , where hw  w is the domain of the finite element mesh:

 h (x, t)  N*  x  Ψ t  ,   x    Ψ*  N  x  ,

(88)

uh  x, t   N*u  x   U t 

(89)

where N*  x  is the row vector, N*u  x  is the matrix of shape functions for the finite element approximation of fields  in volume h w and u on boundary ws , respectively;

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196

A. V. Nasedkin

Ψ  t  , U ( t ) are the global vectors of nodal fluid velocity potentials and nodal solid displacements, respectively. Substituting (88), (89) into the problem (84) with (85)–(87) for h w , we obtain the finite element equation for acoustic medium

R*u  U  R*u  U  M Ψ  C  Ψ  K  Ψ  0 ,

(89)

where

R*u   w 

R*u   wR*u , M 

w 2 w

c



hw

hws

N N* d  

C   wK 

 w2 Z



hwi

2 w2 w Z



hwi

N N* d  ,

N  nw  N*u  d  ,

N N* d  ,

(90)

(91)

K   w 

hw

S*  S d  , S  N* . (92)

Here on the boundary hw  hw , we introduce the boundaries h wf , h wc , h wi ,

h ws , which are approximations of the corresponding boundaries  wf ,  wc , wi , ws . In (90)–(92) matrices M , C , K and R u are assembled from element matrices Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

ek ek ek , C , K and R uek , defined by (90)–(92) with integration by element volume  ek M hw

and corresponding boundaries. We then subdivide the acoustic elements into three types: usual acoustic element, acoustic element with fluid-structure coupling boundaries ek h ws and acoustic elements with impedance boundaries  ek h wi . For the finite elements adjoining to boundary h ws with fluid-structure coupling, we calculate the element matrices R uek and

R uek with integration by boundary ek h ws . However, these acoustic elements have both acoustic degrees of freedom Ψ and structural degrees of freedom U on ek h ws . For acoustic finite elements having the impedance boundary  ek h wi , we can calculate the surface integral on ek ek ek h wi for the element matrices M and C .

The global matrices M , C , K are symmetrical; and M is always positive definite; C is positive definite if boundary h wi exists or if hwf   ; K is positive definite if hwf   .

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 197

2.5. Finite Element Modelling for Coupled Piezoelectric and Acoustic Media Let us assume that the piezoelectric body    p has a fluid-structure coupling boundary that coincides with boundary ws from the previous section. Then, for the finite element approximation (52) in h 

k

ek , we have for a transient problem system similar

to (54), (55)

Muu  U  Cuu  U  Kuu  U  Ku  Φ  Fu ,

(93)

 d K*u  U  K*u  U  K  Φ   d F e  F e  F ,

(94)

In (93), vector Fu contains both the structural vector Fu and an additional vector from the stress n  σ  ns  σs on the boundary ws with normal n s :

Fu  Fu  Fu ,

Fu  

hws

Nu   ns  σs  d 

(95)

Because n s   n w , then by using (82), (88), (90) we can evaluate vector Fu from (95) through an unknown nodal acoustic velocity potential Φ on ws :

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Fu  Ru  Ψ  Ru  Ψ .

(96)

With results from (93) with (95), (96); (94); (89), we obtain the following finite element system for piezoelectric solid interacting with acoustic medium:

Muu  U  Cuu  U  Kuu  U  Ku  Φ  Ru  Ψ  Ru  Ψ  Fu ,

(97)

 d K*u  U  K*u  U  K  Φ   d F e  F e  F ,

(98)

R*u  U  R*u  U  M Ψ  C  Ψ  K  Ψ  0 .

(99)

By using a common vector of an unknown degree of freedom

a  {U, Φ, Ψ

(100)

system (97)-(99) can be represented in the general form:

Ma  Ca  K a  F ,

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(101)

198

A. V. Nasedkin

where

Muu  M 0  R*u

 Cuu 0 R u    0 0  , C   d K *u * 0 M   R u

 K uu  K  K *u  0

K u K  0

0   0  K 

0 R u   0 0 , 0 C 

Fu     F   d F e  F e  F   ,   0  

(102)

(103)

Here we have multiplied Equation (98) by ( 1 ). For the transient problem, we need to add the following initial conditions:

U(0)  U* , U(0)  U* ,

Ψ(0)  Ψ* ,

Ψ(0)  Ψ* ,

(104)

where U* , U* are the initial vector-values (56); and Ψ* , Ψ* are the initial values for acoustic degrees of freedom, usually Ψ*  0 , Ψ*  0 .

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Equations (101) – (103) are extensions of the FEM systems used in finite element program packages ANSYS [9] and PZFlex [7, 8]. These equations usually lead to forms [5, 8], where acoustic dissipation, electric damping coefficient are absent ( b  0 ,  d  0 ) and no boundary conditions exist for punches. Further, we use symmetric algorithms with saddle matrices for solving Equations (101) – (103) for other problems. Note, however, that (for example, in ANSYS), as the main acoustic unknown function often accompanies acoustic pressure P instead of acoustic velocity potential Ψ . Since in classical acoustics with b  0 in (69), the equation P  0Ψ is applied, after derivation (99) by time using b  0 , the received equation may be recorded using nodal unknown vectors U and P . We can also represent the equation (97) using vector P instead of vector Ψ . However, the resulting order of time derivative at vectors multiplying by matrix R (*) u in (97) is lower, while in (99) it is higher, leading to additional failures of symmetry. Therefore, the coupled system (97)–(99) is preferable.

3. ALGORITHMS FOR PIEZOELECTRIC FINITE ELEMENT ANALYSIS 3.1. The Newmark Scheme for Solving Transient Problems It is well known that methods using direct time integration are more general than others. We will use the Newmark method for integrating Cauchy acoustopiezoelectric transient

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 199 problem (97)–(99) or (101) in a formulation in which velocities and accelerations in the time layers are not explicitly given. The conventional Newmark scheme is based on the expansion of vector functions a i 1  a( ti 1 ) , a i 1  a( ti 1 ) , where a  {U, Φ, Ψ} for acoustopiezoelectric problem,

a  {U, Φ} for piezoelectric problem, a  {U, Ψ} for coupled structural-acoustic problem and so on ( ti  i ,   t is the time step size)

1 aip1  ai   ai  (   ) 2ai , 2

ai1  aip1   2ai1

(105)

aip1  ai  (1   ) ai ,

ai1  aip1   ai1

(106)

where  and  are the parameters of the Newmark method. We introduce the averaging operator Yi

Yi b  k 0 k bi 1k , 2

b  a, Fu , F e , F

(107)

where

1   1  2 , 2   2   ,

bi  b(ti ) , 0   ,  1  1/ 2   ,  2  1/ 2   . The following lemma holds. Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Lemma 3.1. Suppose the quantities a i , a i , a i , aip and aip are connected by the relations (105)–(107) for all  i  N . Then, the following equalities hold:

Yia p   1ai   2ai1 ,

Yia p  (ai  ai1 ) / ,

(108)

Yia  ( ai1  (2  1)ai  (1   )ai1 ) / ,

(109)

Yia  (ai1  2ai  ai 1 ) /  2 .

(110)

It is also easy to show that the quantities Ya and Ya may be expressed in terms of i i using the following formulae: Yi a , Yi a p and Ya i p

Yi a 

1



2

(Yi a  Yi a p ) ,

Yi a 

 (Yi a  Yi a p )  Yi a p , 

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(111)

200

A. V. Nasedkin We act on (101), written for the instant of time ti , the averaging operator Yi (107).

When the expressions (111) are taken into account, we obtain the following systems of linear algebraic equations for each time layer:

Keff  ai1  Fieff 1 (Yi Fu , Yi F e , Yi F e , Yi F , ai , ai 1 ) ,

(112)

where

K eff

eff  K uu K u  * eff   K u K  K ueff * 0 

Fueff,i 1  

1



(113)

 C  K ,   u,  ,    1    eff  R u  R u , K  (1  d ) 1 K 2   

eff K 

eff K

 Fueff, i 1  K ueff    eff  0  , Fieff 1   F , i 1  eff  Feff  K   , i 1  

1

2

1



M 

Yi Fu 

1



2

Muu  (2Ui  Ui 1 ) 

Cuu  ((2  1)Ui  (1   )Ui 1 ) 

1



1

 2

R u  (2Ψi  Ψi 1 ) 

R u  ((2  1) Ψi  (1   ) Ψi 1 ) 

1    Ui  2 Ui 1 )  K u  ( 1 Φi  2 Φi 1 ) ,     1 1     (1  d )F eff,i 1  Yi F e  d Yi F e  Yi F  d K *u  ((2  1)Ui  (1   )Ui 1 )            K *u  ( 1 Ui  2 Ui 1 )  K   ( 1 Φi  2 Φi 1 ) ,    

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 K uu  (

Feff,i 1  

1



1



2

R*u  (2Ui  Ui 1 ) 

1

 2

M  (2Ψi  Ψi 1 ) 

R*u  ((2  1)Ui  (1   )Ui 1 )   K  (

1



C  ((2  1) Ψi  (1   ) Ψi 1 ) 

1  Ψi  2 Ψi 1 ) .  

Thus, we have a step-by-step scheme in which for time layer ti 1 it is necessary to solve the system of linear algebraic equations (112) with a symmetric effective stiffness matrix of

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 201 *

saddle structure. This matrix can be factorized using the LDL -factorization method, and only the systems of linear algebraic equations with lower and upper triangular matrices can be solved in each time layer. According to the lemma, the Newmark scheme presented here is mathematically equivalent to the usual Newmark scheme with velocities and accelerations [31]. Consequently, the scheme is absolutely stable when

  (1/ 2   )2 / 4 ;   1/ 2 ; and

when   1/ 4 ;   1/ 2 , it does not have an approximation viscosity. However, the Newmark scheme (112), (113) does not explicitly use velocities and accelerations, and this makes it preferable in the case of the acostopiezoelectric problems, as considered here, when there are no accelerations (and there are no velocities with  d  0 ) of the electric potential in the equations. Finally, we describe some generalized bordering methods for equations with saddle matrices [35]. Definition 3.1. For matrices B and G ordering ( n  n ) and ( m  m ) respectively, we shall call the matrix K of the saddle type (n, m) matrix with respect to block decomposition (  ,   ), if matrix K has the following block form:

B K * H

H , G 

(114)

and B , G are the positive definite matrices. Then the following lemmas and theorem hold: Lemma 3.2. For the saddle type (n, m) matrix K we have sign (det K)  (1)m .

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Lemma 3.3. Let the saddle type (n, m) matrix K ( n  m  N ) is the edging result of the

n1 matrix M of type (N-1) by the vector u  R and the quantity s  R :

M u K * . u s 

(115)

Then, if s  0 , then the matrix M is the saddle matrix of type ( n  1 , m) and

s  u*  M 1  u ; and if s  0 , then the matrix M is the saddle matrix of type (n, m  1 ) * 1 and s  u  M  u . * Theorem 3.1. The saddle type (n, m) matrix K has one and only one LDL matrix decomposition:

K  L  D  L* ,

I D n 0

0  , I m 

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(116)

202

A. V. Nasedkin

where L is a low triangular matrix with positive diagonal elements; I n , I m are the identity matrices of type n and m, respectively. In result for saddle matrix K we obtain the generalized bordering method for * determination the LDL -factorization of saddle matrix K .

N  1 have the LDL* -factorization M  LM  DM  L*M . Then, the saddle matrix K (115) of dimension N will have the Let the saddle matrix M of dimension

* following LDL -factorization:

L K   M* w

0  D M  z   0

0   L*M  d   0

w , z

(117)

w  DM1  LM1  u , z  | s  w*  DM  w | , d  sign(s  w*  DM  w)

(118)

*

The generalized bordering method (116)–(118) for determining the LDL -factorization of saddle matrix K has been implemented in the finite element analysis package ACELAN, with the Newmark method for integrating the Cauchy acoustopiezoelectric transient problem (97)–(99) or (101).

3.2. Harmonic and Static Finite Element Analysis For harmonic analysis, we have the steady-state oscillations U  U exp[ jt ] ,

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Φ  Φexp[ jt ] , Ψ  Ψ exp[ jt ] with Fu  Fu exp[ jt ] , F  F exp[ jt ] . In this case, from (100)–(103), we obtain a system of integral relations for the amplitude functions and values:

(2M  j C  K)  a  F ,

a  {U, Φ, Ψ

(119)

where

Muu  M 0  R*u  K uu  K   K *u    0

 Cuu 0 R u    0 0  , C 0  R*u 0 M  K u 1  K  (1  j d ) 0

0 R u   0 0 , 0 C 

(120)

  0  Fu    1   0  , F   F e  F   , (121)  (1  j d )      K  0

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 203 System (119) is a system of linear algebraic equations with a symmetric matrix and complex arithmetic. The matrices M and C have a saddle structure, and the matrix K is the symmetric matrix with complex coefficients. Its structure is similar to the saddle structure. For static problems, we do not have coupled acoustic and piezoelectric media, and the static finite element analysis consists of the following system:

K  a  F , a  {U, Φ ,

(122)

where

 K uu K * K u

K u  , K 

Fu   F ,   F F    e  

(123)

with the symmetric saddle matrix K .

3.3. Some Algorithms for Modal Analysis As we note in section 2.2, the electric resonance frequencies f rk  rk /(2 ) and the electric antiresonance frequencies fak  ak /(2 ) are the most important characteristics of

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piezoelectric devices. The electric resonance and antiresonance frequencies are the eigenfrequencies of piezoelectric body and are calculated without taking into account the damping effect and the acoustic loads. Hence, these frequencies can be found by the finite element method from the solution of the generalized eigenvalue problem:

K  a   2M  a ,

(124)

where

M M   uu  0

0  K uu  K , K * 0  u

K u  U  , a  .  K  Φ 

(125)

Note that we have used the denotation M uu for the matrix Muu from section 2. Here the matrices K u

and K in (125) differ for electric resonances and

antiresonances. If the matrix K and the vector Φ for electric resonance frequencies are represented in the block form

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204

A. V. Nasedkin

 K uu  K  K uc*  K us* 

K uc cc K  cs* K

K us  cs  K  , ss K 

Φ  Φ  c,  Φs 

(126)

the matrix K and the vector Φ for electric antiresonance frequencies will have the form

K K   cuu* K u

K uc  , cc  K 

Φ  Φc .

(127)

Here, in vector Φ in (126), we extract separately the degrees of freedom Φ s for which, when finding electric antiresonance frequencies, we assume that the electric charges in the corresponding nodes are equal to zero; and when finding the electric resonance frequencies, we assume that the electric potentials are equal to zero – Φs  0 . Thus, to determine the electric resonance and antiresonance frequencies, we need to solve problem (124), (125) twice with matrices K in the forms (124) and (125). The mathematical investigation of the continual forms of eigenvalue problems (124)– (126) and (124), (125), (127) was carried out in section 2.2 (see also [30]). The obtained results transfer to the digitized finite element problems. Thus, we have the following properties. The matrices M and K symmetric: M uu  0 ; K  0 ; Kuu  0 ; K  0 ; cc ss K  0 ; K  0 , where the inequalities A  B and A  B for the matrices denote that

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the matrix ( A  B ) is positive definite or positive semi definite respectively. In this case we assume that at least one constraint is imposed on the electric potential field  which eliminates its indefiniteness within the arbitrary constant. Problems (124)–(126) or (124), (125), (127) can be represented in the form

 Muu  U  Kuu  U ,

  2 ,

(128)

where 1 Kuu  Kuu  Ku  K  K*u ,

1 Φ  K  K*u  U .

(129)

r a The following inequalities are satisfied for the matrices K uu and K uu for the problem of a r a r resonance and antiresonance frequencies: Kuu , where K uu and K uu are obtained  Kuu

from (129), (127) and (129), (126) respectively. The eigenvalues

 rk  rk2 and  ak  ak2 ( k  1, 2,..., n ; n is the order of matrices

M uu and K uu ) are real and non-negative. The eigenvectors corresponding to them, which n we will denote by Wrk and Wak , form a basis in R , where these vectors can be chosen

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 205 orthonormal with respect to the mass matrix M uu and orthogonal with respect to the stiffness matrix K uu

W k , W m  W*k  Muu  W m  km ,

(130)

W*k  Kuu  W m  2mkm ,   r, a . If the electric resonance frequencies

(131)

rk and antiresonance frequencies ak are

numbered by increasing order, we have the inequalities

rk2  ak2 , k  1, 2,..., n . Thus, the coupled eigenvalue problems (124), (125) with respect to the triple of unknowns {  , U , Φ } are in fact the generalized eigenvalue problems (128), (129) with respect to the pairs {  , U }. Changing from (124), (125) to problem (128), (129), we can discuss the procedure of static condensation by eliminating the degrees of freedom of Φ . If, when realizing this procedure in practice, we form the matrix K uu explicitly, the properties of the sparseness of the matrices of finite element method will be lost. With such a strategy for solving generalized eigenvalue problems, we can use methods that transform the matrix

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K uu , such as the Householder – Bisection – Inverse (HBI) iteration method [9, 31]. However, the algorithms that use block forms of the matrices and preserve the sparseness structure are more attractive. We will show that, to solve eigenvalue problem (128), (129), we can effectively adopt modern methods of solving generalized eigenvalue problems for large sparse matrices. For particular problems, these methods require the user to employ procedures of multiplying a matrix by a vector, the addition of matrices and the solution of a system of linear algebraic equations. Depending on the method employed, and also on the problems of searching for groups of eigenvalues with some extremality properties, eigenvalue problem (128) can be modified to one of the following fundamental forms: 1 (I) AM  U   U , AM  Muu  A ,

(132)

(Is)

As  y   y ,

(II)

AM1  U   U , AM1  A1  Muu ,

(134)

(IIs)

As 1  y   y ,

(135)

As  LM1  A  LMT ,

As 1  LTM  A1  LM ,

where

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(133)

206

A. V. Nasedkin

y  LTM  U ,

Muu  LM  L*M ,

A  Kuu   Muu

(136)

LM is the Cholesky factor of the matrix M uu ,  is the value of the shear and

     ,   (   )1 . The matrices A s and A s 1 of problems (Is) and (IIs) are symmetrical, while the methods of solving these problems maintain the usual orthogonality of calculated eigenvectors y . The eigenvectors U  W k obtained from (136) then turn out to be orthogonal in a sense of the scalar product from (131), that is, they are orthogonal with respect to the matrix of masses. The matrices A M and A M1 of problems (I) and (II) can be regarded as symmetrical n operators in R space with a scalar product from (131)

AM1  a, b  a, AM1  b , a, b  Rn

(137)

Therefore, for problems (I) and (II) , the orthogonality of the vectors U is usually maintained in the sense of the scalar product from (130). Thus, problems (I) , (Is) or (II) ,

(IIs) and methods for solving them are very similar; in fact, they differ solely by the scalar product employed. In this connection, we will only consider problems (I) and (II) below. In order to solve problem (I) by iterative methods for large sparse matrices, we need to carry out the procedure of multiplying matrix A M by the iterated vector U and multiplying

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the matrix M uu by a vector to calculate the scalar product from (130). To do this, at the preparatory stage, we must form the matrix

Auu  Kuu   Muu ,

(138)

and obtain Cholesky factors of matrices M uu and K ( K  L  L* )

M uu  L M ,

K  L .

(139)

Further, the algorithm for calculating z  AM  U for block matrices, as follows from (133), (129), (138) and (139), can be realized as follows:

x1 : K*u  U Solve L  x2  x1 for x 2 Solve L*  x1  x2 for x1

v1 : Ku  x1

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 207

v2 : A uu  U

(140)

v1 : v1  v 2 Solve LM  v2  v1 for v 2 Solve L*M  z  v2 for z To solve problem (II) , we need a procedure for calculating the vector z  AM1  U or

z  AM1  v , v  Muu  U for the iterated vector U . Thus, here, to determine the vector z we must solve the system of linear algebraic equations:

 A uu G  K   M   *  K u

K u  ,G K  

z  v       x   0 

(141)

for the vector {z, x} , or the system of linear algebraic equations

A  z  v

(142)

for the vector z . *

The system of linear algebraic equations (141) can be solved using LDL -factorization of the matrix G and subsequent separation of the vector z from the vector of the solutions.

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This method does not require the formulation of blocks of the matrix K . An alternative approach is to use iterative methods to solve system of linear algebraic equations (141), or (142), for which multiplication of G or A by vectors is required. For the block approach for system of linear algebraic equations (142), this operation is carried out in the first six steps of the algorithm (140). Note that the methods for the solution of problem (II) or (IIs) have the inverse iterations in at their foundation and consequently are preferable for use in searching for groups with the lowest eigenfrequencies, i.e., those problems most frequently met with in practice [24].

3.4. Some Algorithmic Features of Practical Implementation Here we consider some techniques to account for specific main boundary conditions in finite element piezoelectric analysis. For simplicity, we limit our examination to homogeneous boundary conditions for modal analysis. First we note that approximation (52) should satisfy the principal boundary conditions (25)–(28) on  huj , h  j . Let

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208

A. V. Nasedkin

uh (x, t )  N*u (x)  U(t) ,

h (x, t )  N* (x)  Φ(t ) ,

(143)

be the finite element representations for functions constructed without taking into account the principal boundary conditions. Let us consider in detail the various types of the principal boundary conditions. The boundary conditions (26) and (28) determine the equalities to zero for values of some degrees of freedom on  huj , j  J r ; h  j , j  J es . The boundary conditions (27) for each electrode h  j , j  J eo connect some degrees of freedom of electric potential  k with one value  l

 k   l , k : xk h j ,

xl h j ,

j  J eo .

(144)

Then, by the boundary conditions (25) in nodes xk huj , j  J p of finite element mesh, the normal components unk of displacement vector are connected by the linear ( j) equation with the normal component unO of displacement vector in the node coincident with

the origin O( j ) of the punch local coordinate system O( j )1( j )2( j )3( j ) and with the punch rotation angles

 (1j ) and  (2j )

( j) unk  unO  2(kj )1( j )  1(kj )2( j )

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where

k : xk huj ,

jJp

(145)

1(kj ) and 2(kj ) are the coordinates of the node x k in the local coordinate system

O( j )1( j )2( j )3( j ) for plane region  huj . All types of principal boundary conditions may be written in common general vector form of the constraint equations

U  Q*n  Un ,

Un  Gu  U ,

Φ  G  Φ ,

(146) (147)

where Q*n is the matrix determining orthogonal rotations of nodal coordinate systems,

Un  {Uc , Θ} , U c is the vector of independent degrees of freedom for mechanical displacements, Θ is the vector of independent degrees of freedom for punch rotation angles,

Φ is the vector of independent degrees of freedom for electric potentials, G u and G are the transform matrices setting the constraints for degrees of freedom. The substitution (146), (147) in (143) defines uh and h as the functions that satisfy principal boundary conditions (25)–(28)

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 209

uh (x, t )  N*u (x)  U(t) ,

h (x, t )  N* (x)  Φ(t ) ,

(148)

where

N*u (x)  N*u (x)  Q*n  Gu , N* (x)  N* (x)  G

(149)

As a result, formulas (148), (149) give suitable approximations for the functions from

V , Vht m , corresponding to (52). In the finite element representations for weighting t hul

functions v(x)  Vhul ,

 (x)  Vhm , we include the same shape functions N*u (x) , N* (x)

specified in (149). Note that in practice, the FEM matrices M uu , K uu , K u , and so on, are obtained by various special transformations of the expanded finite element matrices initially constructed without taking into account the principal boundary conditions. In these cases, the transform matrices G u and G from (146), (147) explicitly are not under construction. Satisfaction of the condition U k  0 for some components of a nodal degree of freedom for displacement requires a reduction of matrices by deleting the kth row and column in the matrices M uu and K uu and kth row (column) in matrix K u ( K *u ). For electric potential changes, the condition k  0 does not require a reduction of matrices. as degrees of freedom  k are "massless" and their number does not change the dimension of eigenvalue problem (124). Here, to account for the condition k  0 , it is

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enough for null corresponding kth lines and rows in matrixes K and K *u ( K u ) to have a diagonal element K kk without change. Coupling of degrees of freedom for electric potential by (144) can be carried out by a summation of the kth rows and columns of the matrix K in lth row and column and kth row (column) of matrices K *u ( K u ) in lth row (column). Here, the reduction of matrices is not obligatory, as in the whole system (124), (125) instead of kth rows of the equations (124) with K after their summation, it is possible to write down the constraint equations (144) in the form K kk l  K kk k  0 ; and for conservation of the system structure to the lth row, it is possible to add the equality

 K k

kk

l  k K kk k  0

which is the consequence of the other equalities. The account of the principal boundary conditions (145) for plane punches can be carried out as follows: Let a separate punch in the region  huj contain K j nodes of the finite

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210

A. V. Nasedkin

element model, and let the number of rows and columns of the expanded matrices M uu and

K uu , corresponding to degrees of freedom unk for the nodes contoured by  huj , run over the set of indexes J j , | J j | K j . To account for constraint (145), instead of rows and columns of matrices M uu and K uu with numbers from J j , it is necessary to make three new rows ( j) and columns for degrees of freedom unO ,

 (1j ) and  (2j ) . The rows and columns,

( j) corresponding to unO , are obtained as a result of the summation of rows and columns with

numbers from the set J j . The next two rows and columns, corresponding to

 (1j ) and

 (2j ) , are also obtained by summation of rows and columns with numbers from J j , but with preliminary multiplication to the factors

2(kj ) and ( 1(kj ) ). Corresponding rows (columns) of

the matrices K u ( K *u ) require similar transformations. After carrying out all these transformations, in order to receive diagonal components of the mass matrix M uu , ( j) corresponding degrees of freedom unO ,

 (1j ) and  (2j ) , it is necessary to add the inertial

characteristics m (0j ) , m(1 j )  J  ( j ) ( j ) and m(2j )  J  ( j ) ( j ) . We note, that all of these methods 2

2

1

1

of account for the principal boundary conditions maintain the structure of the finite element matrices from (124), (125).

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4. FINITE ELEMENT MODELLING OF PIEZOELECTRIC DEVICES WITH GYRATION AND TEMPERATURE EFFECTS 4.1. Introduction In this section, as an example of piezoelectric devices with rotation and temperature effects, we investigate one type of piezoelectric gyroscope. Solid piezoelectric vibratory gyroscopes have recently received fairly wide distribution relative to other gyroscopic devices. Piezoelectric gyroscopes usually operate on resonance frequencies and register rotation or gyration frequency by means of Coriolis forces. In section 4.2, we consider one piezoelectric gyroscope in the form of a ferroelectric, partially polarized plate with divided electrodes. This gyroscope, proposed earlier in [36], works on thickness-shear resonance frequencies and on the principle of ― energy-trapped‖ vibrations in the interelectrode region [37]. Finite element method (FEM), as a basic tool for analysis of piezoelectric devices, was applied to solid gyroscope calculations in [38-40]. Current research in ongoing investigations [40] of modelling of vibratory gyroscopes functioning suggests that FEM may be applied along with the technique of solution decomposition by small parameter of the relation of the rotation frequency to the resonance frequency. As distinct from [40], another schema of electrode inclusion is considered, and by a technique described in [41], external electric circuit elements are taken into account. Stepby-step gyroscope analysis using FEM techniques is applied, two- and three-dimensional

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Some Finite Element Methods and Algorithms for Acoustopiezoelectric Problems 211 problems of harmonic vibrations are solved, and the rotation-induced electric potential is calculated. Thus, at various steps of the analysis, the resonance frequencies of the piezoelectric plate, which can be determined numerically by FEM and by analytical approaches [42], have vital importance.

4.2. Analysis of the Vibratory Gyroscope with Rotation Effects A vibratory gyroscope in the form of ferroelectric rectangular plate L0 x W0 with thickness 2h has symmetric relative to center area 2 L x 2W , polarized by thickness 2h is considered. The part of the plate that is situated outside of the area 2 L x 2W is not polarized. On the top surface of the polarized region, three electrodes are located, two of which ( 1 and

2 ) are parallel one another, and the third electrode 3 is perpendicular to the others (Figure 4.1). There are no electrodes on the bottom plate side, and the technological electrodes for polarization have been removed. Relating the piezoelectric plate to the Cartesian coordinate system Ox1 x2 x3 , we directed the axis Ox 3 along the thickness and arranged the plane x1 x2

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in its medial plane. The plate is considered fixed through the whole external contour, with average points x3  0 .

Figure 4.1. Top view of piezoelectric vibratory gyroscope.

While one electrode 3 is fed the harmoniously changing frequency f electric potential

  V0 exp(i2 ft) , the two other electrodes ( 1 and 2 ) are grounded through the resistance R. With this electric input scheme, the potentials on the electrodes 1 and 2 are considered as constants, but with unknown values

  Vj  const j (t) , x  j , j  1, 2 .

Additional equations of the electric circuit connect the potential values V j and the electric charges Q j on the electrodes 1 and 2 through the resistance R. As a result of the input potential difference between the pair of electrodes 1 , 2 and the electrode 3 , the electrical field E is formed, whose direction appears mainly

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212

A. V. Nasedkin

perpendicular to the direction of the polarization vector P. Under the action of the field E, the points of the plate between electrodes 1 , 2 and 3 , and also beneath them, get mainly the displacement component u2 along the direction of the field, which is perpendicular to the thickness and the vector P, that is, the vibration can be considered as thickness-shear. As a consequence of the plate‘ partial polarization, the electrodes‘ disposition and excitation type, the vibrations are localized in the space under the electrodes. This localization of motion can be termed an ― energy trap‖ [36, 37], and it is here that the energy is trapped in the thicknessshear vibrations. For the numerical solution of harmonic problems for the vibratory gyroscope, we used ANSYS. The input data were the same for all calculations: V0  1 (B); the material properties for

piezoceramics

с12E  7.7 1010

PZT-35 (N/m2);

  7.58  103 с13E  7.6 1010

(kg/m3);

с11E  14.4 1010

(N/m2);

(N/m2);

E с33  12.5 1010

(N/m2);

E с44  3.25 1010 (N/m2); e31  2.4 (C/m2); e33  15.0 (C/m2); e15  9.9 (C/m2);

11S  6690 ; 33S  6000 (  0 is the dielectric permittivity for vacuum); the material properties

for 10

E  8.85  10

non-polarized

ceramics

  7.58 103

(kg/m3);

Young

modulus

(N/m ); Poisson coefficient   0.34 ; dielectric permittivity   870 0 ; 2

total Q-factor Q  650 ; external electric resistance R  200 (Ohm); geometric dimensions – L0  25 (mm); W0  25 (mm); 2 L 10 (mm); 2W  10 (mm); l1  2.5 (mm);

w1  3.5 (mm); w3  1.5 (mm); 2h=1 (mm).

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Vibratory gyroscope acts as an angular velocity sensor, as with its rotation around the axis Oz a potential difference exists between electrodes 1 and 2 , which can be easily measured. More precisely, the registration of the rotation Ω  e3 is based on the use of Coriolis forces F  2m Ω  u , where m is the mass, u is the displacement vector. Note that standard FE analysis software packages usually provide no direct tools to account for rotation of the vibratory gyroscope. However, [40] suggests special techniques based on the application of FE model for the rotating and vibratory gyroscope in the conditions, where the frequency of its vibrations   2 f is much greater than the frequency of rotation  , that is, when    . A similar approach is used earlier in [43]. According to the results [44], the equations of motion and quasielectrostatics for the piezoelectric medium allowing for rotation in absolute coordinate system may be written in the form:

 σ(u, )   (u,tt  d u,t  2Ω  u,t  Ω  (Ω  u)) ,   D( u,  )  0 ,

(150)

where  d is the damping coefficient, and other symbols are standard for the theory of electroelasticity. In the case of harmonic vibrations u  u exp(i t ) ,    exp(i t ) from (150), we have:

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 σ(u, )   (2u  id u  2i Ω  u  Ω  (Ω  u)) ,   D( u,  )  0 (151) with Ω   e 3 ,    . Then, it is possible to introduce the small parameter    /  and to present the displacement vector u and the potential  as the expansions on the small parameter:

~  u   u  O( 2 ) , u 0 1

~   0   1  O( 2 ) ,

   / .

(152)

Substituting (152) in (151), for the zero-order approximation u 0 ,  0 we obtain the system

 D(u0 ,0 )  0 ,

(153)

 σ(u1,1 )   (2u1  id u1  2i2e3  u0 ) ;  D(u1, 1 )  0 .

(154)

 σ(u0 ,0 )   (2u0  id u0 ) , and for the first approximation u1 ,

1 we get the system

It is obvious that boundary conditions also can be determined separately for zero and for the first approximation. Thus, for zero approximation, the electric boundary conditions on the electrodes 1 , 2 and 3 will have the form

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 0 V0 , x 3 ,

0  V j , x  j ; j  1, 2 ,

Vj  iQ j (u0 ,0 )R

(155)

Vj  iQ j (u1,1 )R

(156)

and for the first approximation

 1 0 , x 3 ,

1  V j , x  j ; j  1, 2 ,

Whereas, generally speaking the unknown values of the electric potential V j in (155) and (156) are different, Q j (ul ,l )  



j

D3 (ul ,l ) dx1dx2 is the electric charge on the

electrode  j , and D3 is the component of the dielectric displacement vector. Thus, for zero approximation, it is necessary to solve the problem for u 0 ,  0 with Equations (153) and boundary conditions (155) about harmonic vibration of the plate without taking into account the rotation around the axis x3 . The given problem is solved by FEM, and as a result, the displacement field is determined for all nodes of the finite element grid. Further, for the same mesh, it is necessary to solve the problem for u1 ,  1 with Equations (154) and boundary conditions (156). Note that the resulting problem is an ordinary harmonic

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214

A. V. Nasedkin

problem for u1 ,  1 , but with the body forces f  2i2e3  u0 . Then, for the conventional FE equations of electroelasticity

2Muu  U1  i Cuu  U1  Kuu  U1  Ku  Φ1  Fu ,

(157)

K*u  U1  K  Φ1  F ,

(158)

In the nodal body force vector Fu , it is necessary to take into account the contribution of the body forces f. After simple computations, for element vectors of the nodal forces Fue we find:

Fue1k  2i2mkle U02l ,

Fue2k  2i2mkle U01l ,

where U 01l , U 02l are the displacement components in the node with the number l on the axes

x1 and x 2 respectively, found at zero approximation; mkle are the components of the element masse matrixes. To determine the working resonance frequency, we find initially, for some frequency ranges, the gain-frequency characteristic of the electric impedance Z  V0 /(iQ3 ) during the harmonic vibrations of the plate without taking into account the rotation and the external electric resistance. Figure 4.2 shows the resulting plot of the electric impedance gainfrequency characteristic of the piezoelectric gyroscope. As evident in Figure 4.2, with the selected FE meshing, the electric resonance frequency f r  1.073 MHz, and the

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antiresonance frequency f a  1.083 MHz. In analyzing the distribution of displacement U y in the direction of the electrodes 1 , 2 – 3 for different plate sections, we find the pronounced first forms of thickness-mode vibrations for resonance frequency,, and we observe the maximal displacement in the middle part of the plate, between electrodes 1 , 2 and 3 . It is important to note when the plate is excited by the introduction of the voltage between electrodes 1 and 2 with the free electrode 3 , the resonance frequencies almost coincide with resonance frequencies found for the initial scheme of electrode activation. Thus, the first resonance frequencies of the piezoelectric plate thickness-shear vibrations along axes Ox and O y are almost identical, which is important for efficiency of the rotation registration. Further calculations of the vibratory gyroscope are carried out for the resonance frequency f r  1.073 (MHz) using both the asymptotic approach described above and FEM. For calculations, we use finite elements in the form of the parallelepipeds with eight nodes. For the first thickness-shear frequencies, it has been established that four finite elements along the thickness are enough to ensure adequate accuracy with proportional meshing by length and width. The calculations show that for gyroscope rotation with the frequency

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  10 (°C/sec) between the electrodes 1 and 2 , the voltage is V12  7.147 106 (В), which can be safely registered using electric amplification of the measuring circuit to about 60 db. These results are in good agreement with those obtained experimentally in [36] for the gyroscope with the similar configuration.

Figure 4.2. Gain-frequency characteristic of electric impedance.

In addition to the circuit activation accepted in [40], we also carry out optimization calculations to select sizes for the electrodes 1 , 2 and 3 to ensure the most effective

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vibrations on the thickness-shear resonance frequencies.

4.3. Analysis of The Gyroscope with Temperature Effects As a last step in the vibratory gyroscope calculation, we take into account the effects of temperature. As proposed in [45], the solution of harmonic problems on the dissipative heating requires first solution of the problems (157), (158) for mechanical displacements U and electric potentials Φ . Then, using the obtained field of the mechanical displacements, we calculate U the average dissipation function D as follows:

D  2 (Re U*  Cuu  Re U  Im U*  Cuu  Im U) / 2 . This dissipation function is considered as a thermal source for the thermal problem [46] which then can be solved by FEM. Here, the transient FE thermal problem in standard designation has the form

C  T  K  T  F D and can be integrated in time under the Crank-Nicholson scheme.

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The results of these calculations show that in conditions of plane deformation, the vibratory gyroscope with thickness-shear vibrations on the resonance frequency and convection heat transfer on all boundaries is slightly warmed (about 2 degrees), based on a film coefficient equal to 2.5 W/(m2K) and the piezoceramic conductivities equal to kxx  2.5 , k zz  3.75 W/(m∙K). However, additional thermal calculations are required for the piezotransformers and piezoelectric emitters working in high-temperature fields [47]. Considering the temperature dependence of modules of a piezoelectric body, we can determine new piezoelectric modules for temperatures that differ from those of the oscillation period; and also recalculate and solve harmonic piezoelectric problems (157), (158) as well as problems involving dissipative heating. The result is an iterative computing process similar to those described in [46].

ACKNOWLEDGMENTS This work is partially supported by grants from the Russian Foundation of Basic Research.

REFERENCES [1] [2]

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[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

Lloyd, P.; Redwood, M. J. Acoust. Soc. Amer. 1966, vol. 36, pp 346-361. Shulga, N.A.; Bolkisev, A.M. Vibration of piezoelectric bodies. Kiev: Naukova Dumka, 1990; 1-228. Allik, H.; Hughes, T.J.R. Int. J. Numer. Meth. Eng. 1970, vol. 2, pp 151-157. Nasedkin, A.V.; Skaliukh, A.S.; Soloviev, A.N. Russian Izvestiya, North-Caucas. Region. Ser. Natural Sciences. 2001. Spec. Issue. Math. Modeling. pp 23-25. (in Russian) ATILA. Finite-element code for piezoelectric and magnetostrictive transducer and actuator modeling. V.5.1.1. User's Manual. Lille Cedex (France): ISEN, 1997. Landes, H.; Kaltenbacher, M.; Lerch, R. CAPA Users manual. Department of sensor technology, Friedrich-Alexander-University of Erlanger-Nuremberg, 2000, http://www.lse.unierlanger.de/CAPA/ PZFlex, Explicit time domain, piezoelectric, finite element code. Weidlinger Associates Inc., Los Altos, CA. Wojcik, G.L.; Vaughan, D.K.; Abboud, N.; Mould, J. Proc. IEEE Ultrasonics Symp. 1993. vol. 2, pp 1107-1112. ANSYS. Theory Ref. Rel.11.0. Ed. P. Kothnke. ANSYS Inc. Houston, 2007. COSMOS/M. V.2.0. Advanced Modules Manual. ASTAR. Structural Research and Analysis Corp., 1997. Dokmeci, M.G. Int. J. Eng. Sci. 1980. vol. 18, pp 431-448. Allik, H.; Webman, K.M.; Hunt, J.T. J. Acoust. Soc. Amer. 1974. vol. 56, pp 17821791. Boucher, D.; Lagier, M.; Maerfeld C. IEEE Trans. Sonics Ultrasonics. 1981. vol. SU28, pp 318-330.

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[29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

Challande, P.J. Mec. Theor. Appl. 1988. vol. 7, pp 461-477. Ostergaard, D.F.; Pawlak, T.P. Proc. IEEE Ultrason. Symp. 1986. pp 639-644. Cowdrey, D.R.; Willis, J.R. J. Acoust. Soc. Amer. 1974. vol. 56, pp 94-98. Kagawa, Y. J. Acoust. Soc. Amer. 1971. vol. 49, (part.1), pp 1348-1356. Kagawa, Y.; Yamabuchi, T. IEEE Trans. Sonics Ultrasonics. 1974. vol. SU-21, pp 273280. Lerch, R. IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 1990. vol. 37, pp. 233-247. Naillon, M.; Coursant, R.H.; Besnier, F. Acta Electronica. 1983. vol. 25, pp 341-362. Abboud, N.; Mould, J.; Wojcik, G.; Vaughan, D.; Powell, D.; Murray, V.; MacLean, C. Proc. IEEE Ultrasonics Symp. 1997. pp 895-900. Kaltenbacher M. Numerical Simulation of Mechatronic Sensors and Actuators. Springer, Berlin - Heidelberg - New York, 2004. Akopov, O.N.; Belokon, A.V.; Eremeyev, V.A.; Nadolin, K.A.; Nasedkin, A.V.; Skaliukh, A.S.; Soloviev, A.N. Proc. Int. Conf. "Piezotechnics-99", Rostov-on-Don, Azov, 14-18 sept. 1999. vol. 2. Rostov-on-Don, 1999. pp.241-251. (in Russian) Belokon, A.V.; Eremeyev, V.A.; Nasedkin, A.V.; Soloviev, A.N. J. Applied Math. Mech. (PMM). 2000. vol. 64, pp 367-377. Belokon, A.V.; Nasedkin, A.V.;, Nikitaev, A.V.; Petuchkov, A.L.; Skaliukh, A.S.; Soloviev, A.N. Proc. Int. Conf. "Piezotechnics-2002", Tver, Russia, 14-21 sept. 2002. Tver, 2002. pp 171-179. (in Russian) Belokon, A.V.; Nasedkin, A.V.; Soloviev, A.N. J. Applied Math. Mech. (PMM). 2002. vol. 66, pp 481-490. Vasiltchenko, K.E.; Nasedkin, A.V.; Soloviev, A.N. Computer Technologies. 2005. vol. 10, pp 10-20. (in Russian) Belokon, A.V.; Vorovich, I. I. Some mathematical problems of the theory of electroelastic solids, In: Current problems in the mechanics of deformable media, Izv. Dnepropetr. Gos. Univ., Dnepropetrovsk, 1979. (in Russian) Mikhlin, S. G.; Variational Methods in Mathematical Physics, Pergamon Press, Oxford, 1964. Belokon, A. V.; Nasedkin, A. V. J. Appl. Math. Mech. (PMM), 1996, vol. 60, 145-152. Bathe, K. J.; Wilson, E. L. Numerical Methods in Finite Elements Analysis, PrenticeHall, Englewood Clifs, New Jersey, 1976. Zienkewicz, O. C.; Morgan K. Finite Elements and Approximation, N. Y., J. Wiley and Sons, 1983. Krasilnikov, V. A.; Krylov, V. V. Introduction in Physical Acoustics. Moscow: Mir, 1984. (in Russian) Nasedkin, A. V. Russian Izvestiya, North-Caucas. Region. Ser. Natural Sciences. 1999. No 1. pp 48-51. (in Russian) Akopov, O. N.; Belokon, A. V.; Nadolin, K. A.; Nasedkin, A. V.; Skaliukh, A. S.; Soloviev, A. N. Math. Modeling. 2001. vol 13, pp 51-60 (in Russian) Abe, H.; Yoshida, T.; Watanabe, H., Proc. IEEE Ultrason. Symp., 1998, pp 467-471. Shockley, W.; Curran, D.R.; Koneval, D.J., J. Acoust. Soc. Am., 1967. vol 41 (2), pp 981-993. Kagawa, Y.; Tsuchiya, T.; Kawashima, T., IEEE Trans. Ultrason., Ferroelect. Freq. Control. 1996. vol 43, pp 509-518,.

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[39] Kagawa, Y.; Tsuchiya, T.; Sakai, T., IEEE Trans. Ultrason., Ferroelect. Freq. Control., 2001. vol 48, pp 180-188. [40] Golovnin, V. A.; Daniltseva, V. I.; Matusevitch, B. S.; Nasedkin, A. V.; Fedina E. V., Proc. Int. Conf. “Piezotechnics-2003”, Moscow, Nov. 26-29, 2003, Moscow: MIREA, 2003. pp 248-253 (in Russian) [41] Wang, J. S.; Ostergaard, D. F., Proc. IEEE Ultrasonics Symp., 1999, pp 1105-1108. [42] Kosmodamianskii, A. S.; Storojev, V. I. Dynamic elastic problems for anisotropic media, Kiev, Naukova Dumka, 1985 (in Russian) [43] Yang, J. S., IEEE Ultrasonics Symp., 1996. pp 909-912. [44] Ulitko, A. F. Russian J. Solid Mech. ( Izv. RAN. Mekhanika Tverd. Tela.), 1990. vol 6, pp 55-66. [45] Nasedkin, A. V., Proc. IV Conf. CAD-FEM GmbH Soft. Users (Moscow, Apr. 21-22, 2004), Ed. A. S. Sadskii, M.: Poligon-Press, 2004. pp 311-315. (in Russian) [46] Karnauchov, V. G., Coupled Thermoviscoelastic Problems, Kiev, Naukova Dumka, 1982. (in Russian) [47] Rao, S. S.; Sunar, M., AIAA J., 1993. vol 31, pp 1280-1286.

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In: Piezoceramic Materials and Devices Editor: Ivan A. Parinov, pp. 219-242

ISBN 978-1-60876-459-4 © 2010 Nova Science Publishers, Inc.

Chapter 5

IDENTIFICATION OF EFFECTIVE PROPERTIES OF THE PIEZOCOMPOSITES ON THE BASIS OF FINITE ELEMENT METHOD (FEM) MODELING WITH ACELAN (1)

A. N. Soloviev1(1) and G. D. Vernigora(2)

Department of Material Strength, Don State Technical University2, Rostov-on-Don, Russia, Department of Mathematic Modeling, Southern Federal University,3 Rostov-onDon, Russia and Southern Scientific Center of Russian Academy of Sciences4, (2) Department of Material Strength, Don State Technical University,5 Rostov-on-Don, Russia

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ABSTRACT We developed a method of defining effective elastic and piezoelectric properties of irregular structure composite piezoceramics. The method is based on dynamic equivalence. We examine the oscillations of some meaningful volumes of composite materials and find their resonance and antiresonance frequencies. The calculations are carried out in the finite element ACELAN complex using the calculation module for irregular structure composite materials. This work also considers the estimation algorithm of a full set of composite piezoceramics effective constants on basis of static tasks. We use the equations and results received after the ACELAN calculation of vertical and horizontal displacements and of the potential of meaningful volume of porous ceramics of irregular structure. The results of quantitative experiments are given below.

1

e-mail: [email protected]. 1, Gagarin Square, 344000 Rostov-on-Don, Russia. 3 8a Milchakova Street, 344090 Rostov-on-Don, Russia. 4 41, Chehova Street, 344006 Rostov-on-Don, Russia. 5 1, Gagarin Square, 344000 Rostov-on-Don, Russia. 2

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1. INTRODUCTION System calculation with complex geometric configuration and irregular physical structure is a routine task in science and engineering. Computers allow us to carry out such calculations using approximate numerical methods: the finite element method (FEM) [1, 2] is one of them. In recent decades FEM has been given a leading role and wide application. The use of the method is considerably wide and covers many physical tasks. The method has a range of advantages, which enable it to be used widely. Acoustic Electrical Analysis (ACELAN) [3], being developed in the department of mathematical modeling of SFU, is a specialized finite element complex for solving tasks that simulate the function of piezoelectric devices that may come into contact with acoustic media. The calculations are implemented of all practically meaningful modes (static, harmonic, modal analysis and non-steady impacts) in flat, axisymmetric arrangement and also in generalized cylindrical bodies. Recent developments in material engineering have led to the creation of brand new materials. Piezoelectric and piezomagnetiс materials are widely used in different spheres of science due to their unique physical and mechanical characteristics. Mainly they are used in acoustics, computing, radio electronics and control systems, such as piezoresonators, piezogenerators, piezoelectric transducers and piezoengines. The current level of studies is indicated in survey works by D. Berlincourt, D. Kerran, I. A. Glozman, V. T. Grinchenko, A. F. Ulitko, N. A. Shulga, V. Mason, B. Z. Parton, B. A. Kudryavtsev, B. Jaffe., G. Jaffe, and W. Cook [4  9] Among the new classes of piezoelectics and piezomagnetics are piezocomposites with piezoactive structure elements. The characteristics and behavior of these materials are determined by complex interaction of great number of piezoactive elements, forming the constitutive structure of the material by means of intertwined fields of different physical nature. Piezoactive composites are used in cases when traditional piezoelectrics and piezomagnetics (crystals, ceramics, alloys) do not provide necessary complexes of piezomechanic characteristics (for example, mechanical durability). The possibility of controlling the piezocomposite structure and its optimization clears the way to creation of new piezomaterials with predetermined electric- and magnet-mechanical characteristics. As a result of microlevel interaction of piezoactive structures, composites may gain new macroscale effects that not available in homogenious piezomaterials. Porous piezoceramics have a special rank among these newly-created materials, depending upon their method of production. The production of porous piezoceramics allows various material characteristics, such as volumetric content, size and distribution of pores (surveyed in [10, 11]). Thus, the study of electric, acoustic, elastic, piezoelectric and dielectric properties of porous piezoceramics and detection of their dependencies from pore microstructure, manufacturing method etc. [15, 28] is agermain. Various methods of porous piezoceramics (PZT) manufacturing were surveyed from the microstructure view point, revealing a wide range of volumetric content and a wide distribution of pore size [10]. Three methods are examined here: (a) PZT sintering temperature drop; (b) various methods of founding (beginning with oxyhydroxite with highvolumetric organic content); (c) increasing the volumetric concentration of organic polymer in PZT powder. The foreground of reference [11] is dedicated to pore morphology formation,

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Identification of Effective Properties of the Piezocomposites…

221

using pore size and form distribution control. ― Dry‖ and ― wet‖ methods of pore infusion are surveyed. The dry method implies mechanical mixing of volatile phase and powder. The wet method includes manipulation of suspensions. It is usually noted that using this method makes it possible to improve final morphology and microstructure control with the help of a colloid approach. A range of approaches to porosity formation is surveyed. Porosity formation includes burning of volatile particles, formation of thermally unstable materials, foaming, slip and ribbon casting, and direct consolidation. As mentioned, pore size distribution and microstructure differences in samples result from different manufacturing parameters, which. influence pore microstructure and other physical properties, especially PZT acoustic and piezoelectric effects. The study of electric, acoustic, elastic, piezoelectric and dielectric properties of porous piezoceramics and the detection of their dependencies on pore microstructure, manufacturing method, etc., is also instructive. Reference [12] describes the effects of different pore-forming materials (PFM), including their contents in microstructure and piezoelectric properties of piezoceramics, their percentage of porosity (varying from 5% up to 45%), and whether they are created with the help of PFM such as polymethyl methacrylate (PMMA) and dextrin (the optimal sintering temperature is 1200° C). There appear to be no other meaningful differences in properties of PMMA- and dextrin-derived materials, even if their pore size varies. The decrease of axial piezomodule and dielectric permittivity registers when porosity of the sintered piezoceramic increases. Also, an increase in the hydrostatic piezoelectric voltage index and the hydrostatic Q-factor are observed. Moreover, hydrostatic charge and voltage index appear to increase if the materiala contain 50% of PMMA [13]. Comparative analysis of irregular porosity ceramics and spherical pore ceramics was carried out [14], determining that spherical pore ceramics reveal improved piezoelectric, dielectric and ferroelectric properties (compared to irregular porosity ceramics, where the pore size is irrelevant) due to low stress. However, a drastic drop in electric properties of 35%-piezoceramics (when the pores are formed with stearic acid [15]) is observed due to simple pore interrelationship. Also, a twofold linear decline of acoustic resistance occurs if porosity increases from 3% to 43%, regardless of the kind of pore morphology. As a result, it is determined that the electric PZT properties are closely connected with porosity and pore interrelationship as opposed to the form of the pores. Acoustic resistance also is found to be connected with porosity percentage. The article [16] also lso surveys samples of 40% porous ceramics. These pores are formed with multilayered graphite, and endow the ceramics with anisotropic porosity [16]. Elastic piezoelectric and dielectric properties of samples with homogeneous porosity also are compared to the samples with a higher degree of anisotropy which is infused by increasing the volumetric content of the pores. The samples show a strong difference in receipt porous stepped material in longitudinal and transverse directions due to strong microstructure anisotropy. The final pore distribution and the properties of the porous stepped material are the result of a combination of homogeneous layers. The samples of these materials also show continuous changes in morphology without delamination of the boundaries between layers. Moreover, it may be concluded that the choice of production method can determine the properties of porous stepped material more precisely. Reference [17] surveys porous PMN–PZT ceramics made of sintering pressing. Piezoelectric and dielectric properties of the ceramics obtained are studied as functions of density. The survey indicates that density decreases in the ceramic as PMMA content

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increases, leading to decrease of modulus of elasticity and increase of mechanical compliance. This data points to the fact that matching of acoustic resistance between PMN– PZT ceramics and the medium can be improved provided the ceramics‘ density is decreased. The author also points out that decreased density results in an increase in the hydrostatic coefficient of Q-factor and a decrease in dielectric permittivity. Moreover, PMN–PZT ceramics with low density showed lower dielectric loss compared to the same ceramics of higher density. The survey‘s discussion of microstructure and piezoelectric properties of porous ceramic samples is of particular interest. Ceramics are made with the addition of polymethylmethacrylate (PMM), which forms pores at different temperatures of sintering. The conclusions of Reference [18] indicate that, when surveying 34%-porous ceramic samples an increase of dielectric permittivity, the longitudinal piezoelectric coefficient and hydrostatic coefficient of Q-factor are observed (the sintering temperature is increased from 1050 to 1300 C). The author also points out that the longitudinal piezoelectric coefficient of 34%-porous ceramics is nearly identical to the longitudinal piezoelectric coefficient of 95%PZT ceramics (at a temperature of sintering of 1300 C), whereas hydrostatic coefficient of Q-factor of 34%-porous ceramics is 15 times higher than that of 95%-ceramics. The dielectric permittivity of 34%-porous ceramics is much higher than that of PZT polymer composite at a sintering temperature of 1300 C. Reference [19] surveys PZT polymer composite made by means of hydrocarbon infiltration into porous PZT ceramics. The relative volume of ceramics varies from 68% to 87%. Electric and mechanical properties of the composite obtained are surveyed as functions dependent on ceramics relative volume. It is determined that the density of porous ceramics and composite material decreases if the relative volume of the ceramicis reduced. If the ceramic phase is decreased, the longitudinal piezoelectric modulus and dielectric permittivity of composite material decrease, whereas the hydrostatic strain and Q-factor coefficient increase. The calculation data of the hydrostatic Q-factor coefficient also is presented in the article.

2. CONTINUAL AND FINITE-ELEMENT PROBLEM STATEMENTS 2.1. Continual Statement of Problem of the Acoustic Electric Elasticity Let us examine a piezoelectric transducer , presented by a set of areas

 j   pk ; k

= 1, 2,..., Np; j = k with the properties of piezoelectric materials and a set of areas

 j  em ;

m = 1,2,...,Ne; j = Np + m with the properties of elastic materials. It is

appropriate to describe the physical-mechanical processes taking place in the media and

em

 pk

, with in the framework of piezoelectricity (electric elasticity) and elasticity theory.

We suppose that the following equations and determining correlations are satisfied (piezoelectric medium is

 j   pk )

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Identification of Effective Properties of the Piezocomposites…

223

   dj  j u -   σ  f j ;  pk u

D0,

σ  c Ej  (ε   djε )  eTj  E ;

  e j   (ε   d ε )  э Sj  E ,(2) D dD

ε  (u  uT ) / 2 ;

E   ,

(1)

(3)

where   x, t  is the continuous function of coordinates (density), u(x) is the displacement vector-function,  is the stress tensor, f are the mass forces, D is the electric induction vector,

cEj are the components of the elastic constant tensor,  is the strain tensor, E is the electric field vector,

э Sj are the components of the dielectric permittivity tensor,

  x  is the electric potential function,

 dj ,  dj ,  d

are the nonnegative coefficients of damping.

For the media

 j  em

with pure elastic properties, only stress fields would be

considered. Similar equations (1) – (3) and constitutive relationships are used with neglect electric fields and piezoelectrical connectivity effects.

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Finally, a piezoelectric device can be loaded on acoustic operating media

 j  al ; l

= 1, 2,..., Na; j = l + Np + Ne. For these media we use acoustic equations with linear dissipative effects taken into account as

1 p    v  0 ;  j c 2j  j v    σ ;

v   , σ   p I  b v ,

(4)

(5)

where j is the density equilibrium value; cj is the sound velocity; bj is the dissipative coefficient for the medium

 j  em ; p is the sound pressure; v is the vector of velocity;

 is the velocity potential;  is the strain tensor; I is the identity tensor.

It should be noted that models (1) – (5) have specific methods of damping accounting. A Rayleigh damping accounting model is used for elastic media. This model (1) – (3) is generalized for piezoelectric media. Detailed analysis of the model (1) – (3) can be found in References [20, 21], but Reference [22] provides an analysis of the model (4), (5) for the acoustic medium with dissipation.

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In addition to Equations (1) – (5), it is necessary to define appropriate principal and natural boundary conditions, conditions of field matching along the contact boundaries of different media, impendence conditions for the ‗truncation‘ of acoustic areas and initial conditions of non-stationary problems [22].

2.1.1. FEM Formulation We use FEM in classic Lagrangian formulation to solve dynamic problems of acoustic electric elasticity. We choose coherent finite-element mesh specified in the areas hj, which approach area j. There are unknown field functions u,  and  in this mesh. We approximate them as

u(x, t )  NTu (x)  U(t ) ;  (x, t )  NT  Φ(t ) ;  (x, t )  NT (x)  Ψ(t ) where

Nu

is the shape function matrix for displacement field u; N and

N

, (6)

are the shape

vector-functions for the electric potential fields  and speed potential in acoustic medium , respectively; and U(t), Φ (t), Ψ (t) are the global vectors of the corresponding nodal degrees of freedom.

2.1.2. FEM Systems for Different Types of Problems FEM approximation (6) of the generalized formulations of dynamic problems (1)–(5), including principal and natural boundary conditions, results in the following system of differential equations:

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M  a  C  a  K  a  F ,

(7)

~  M uu 0 Ru   0 0 , M  0 ~T  R  u 0  M  

 C uu 0 Ru    T C    d K u 0 0 ,  RT 0  C   u

0 0

 K uu K u  K   K Tu  K   0 0 

Fu       , F  F   d F ,    0  K    

relatively

vector

of

the

Cuu   j ( dj Muuj   dj K uuj ) ,

of

unknown

where

M uuj

and

(8)

a  U, Φ, Ψ . K uuj are the structural T

Here finite

elements of mass and stiffness matrix. Other elements of the submatrix (7), (8) are described in Reference [22]. FEM system for eigenvalue (eigenfrequency) problems is the following

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Identification of Effective Properties of the Piezocomposites…

  2M  a  K  a  0 ,

Steady-state

(9)

 K uu 0  T  K , K 0  u

M M   uu  0

vibration

225

problems

K u    K  . are

arise

(10) when

~ Fu  Fu (x) exp( jt ) ,

~ F  F (x) exp( jt ) , a  ~ a(x) exp( jt ) . It is easy to derive a system of linear a algebraic equations from (7), (8) relatively of the amplitude vector ~

~ Kc  ~ a  Fc ,  K uuc  K c   K Tu  KT  uc





~ ~ ~ Fc  Fu , F ,0 K u  K  c 0

T

,

(11)

K uc   0 ,  Kc 

(12)

K c   2 M  i C  K ,   u,  ,

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~ K uc   2 Ru  i Ru , K  c 

1 K  (1  i d )

.

2.2. Development of Electrostatic Elements in ACELAN Software for Porous Electric Elastic Media. Finite-Element Modeling of Composite Materials with Irregular Structure, and Visualization of the Control Process of Pore Distribution in Composites with Irregular Structure The ACELAN software‘s modulular architecture allows easy addition of new clusters (triangulators and solvers) and separate use of available. However, existing ACELAN versions have no option to operate with electrostatic finite elements. Thus, one of the goals of this effort is to develop and implement electrostatic elements in the ACELAN software and employ them to model porous media. To end, a new solver was created, which enabled us to deal with problems relating to bodies with just one degree of freedom, namely electric potential. The new solver was developed on the basis of an existing one that helped to solve problems related to stationary vibration. This work presents some of the results obtained using the new solver. It is known that by modeling a porous medium one can choose a representative volume corresponding to regular structure. New, modern composites often have regular or nearly

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A. N. Soloviev and G. D. Vernigora

regular structures in which the heterogeneity scale is relatively small compared with their overall size . Considering a homogeneous porous body, we solved 4 model problems for one pore using the newly developed solver. The upper borders of the construction are free, but lateral borders have electrodes with potential difference.

1 – electric elastic medium 2 – medium modeling pore

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Figure 1. Representative volume of a homogeneous porous body.

In the first modeling problem, the pore is presented as a hole without material. In this case, the electrostatic problem inside the body has no solution and no influence of the field. In the second model, the pore is present as an acoustic medium, like air. In this case, we consider the pore as an electrostatic medium, a continuum, which has nocorresponding properties of elastic and acoustic media. Finally, the pore is present as an uncoupled elastic and electrostatic medium with minimal stiffness, i.e., an elastic body with minimal stiffness, in which the electric component is determined by dielectric permeability. We chose the following 3 nodes in which to study the models: Node 12 is the node disposed in the inner border of the electric elastic medium and the medium modeling the pore (Figure 1). Node 21 is the node disposed in the outer border (Figure 1). Node 20 is the node disposed inside the body examined (Figure 1). The amplitude-frequency characteristics were plotted for all three nodes. The analysis of the characteristics allows one to conclude that the four models exhibit the same behavior at low frequencies (Figure 2). The solution of the steady-state oscillation problem (i.e., the plane problem) is presented in Tables 1 and 2 for selected low  1  and high ( 2 ) frequencies. Both tables contain

information about vertical (u ) and horizontal ( ) displacements, electric potential ( ) , and velocity potential ( ) for node 12. These results allow one to conclude that the values approximately coincide at the low frequencies.

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Identification of Effective Properties of the Piezocomposites… Table 1. Results for frequencies 1 = 4.5103 Node 12

u  10 8

  10 7

 10 2

  10 4

Model 1

0.4029 0 0.3919 0 0.3912 0 0.3903 0

0.3429 0 0.3409 0 0.3403 0 0.3393 0

0.9772 0 0.9773 0 0.9755 0 0.9721 0

0 0 0 0.05366 0 0 0 0

Model 2 Model 3

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Model 4

Figure 2. Amplitude-frequency characteristic of horizontal displacement of node 20.

Table 2. Results for frequencies  2 = 1.584104 Node 12

u  10 8

  10 7

 10 2

  10 4

Model 1

 0.2776 0  0.3475 0  0.3470 0  0.3477 0

0.1257 0 0.2258 0 0.2258 0 0.2263 0

0.2404 0 0.3549 0 0.3549 0 0.3556 0

0 0 0  0.1668 0 0 0 0

Model 2 Model 3 Model 4

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A. N. Soloviev and G. D. Vernigora

3. FINITE-ELEMENT MODELING OF COMPOSITE MATERIALS WITH IRREGULAR STRUCTURE AND VISUALIZATION OF CONTROL PROCESSES OF PORE DISTRIBUTION IN COMPOSITES WITH IRREGULAR STRUCTURE With finite-element modeling of composite materials with irregular structure using ACELAN, a new solver was developed, which helped the authors to consider the problems of steady-state oscillations in the constructions containing multi-component composite materials with irregular structures. By modeling the porous material the solver makes it possible: i) to define pore percentage in composite material ii) to solve various steady-state oscillation problems with the same pore distribution iii) to change the full set of pore material constants using multiplication of material characteristics of solid ceramics on specified reduction coefficient iv) to calculate pore volumetric content

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A test of the developed solver was carried out in examples that model specific problems (Figure 3). One-dimensional piston movements parallel to the vector of preliminary polarization (thickness modes) can be taken as an example of such problems.

Figure 3. Distribution of vertical displacement in first thickness mode of 5%-porous ceramics.

At the solution of the problem of steady-state oscillations for composite constructions by using ACELAN software, it was necessary to control the pore distribution and enable its correction. A visualizing utility was developed by using Microsoft Visual Studio 2005, algorithmic language C# [23], which allowed demonstration of the process of finite-element modeling of the porous composite with irregular structure of pores and control of pore distribution. The main features of this visualization are: o presentation of the number of each finite element,

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Identification of Effective Properties of the Piezocomposites… o o

229

change of the initial distribution of the pores using a computer mouse, calculation of the porosity coefficient.

Figure 4. Program interface.

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The specific functions of the software are described below. 1. – «Choose folder». Enables the user to choose a folder containing the data. The program checks availability of files in the folder. In case of their absence a warning appears. 2. – «Load data». Enables full data loading from the files to RAM to facilitate working with them. Loading is indicated with a progress bar. 3. – «Show body». Enables display of the model porous composite with red elements indicating pores (see Figure 4). 4. – «ZOOM». Increases/decreases the image scale. 5. – «Save pore». Saves types of elements in a file. 6. – «Show numbers». Switches on/off the number of finite elements. 7. – «Status line» Describes the current process and initial volumetric porosity coefficient (after data loading) and new coefficient (after manual destitution change). By clicking on the image, the user can i) Change the element condition to the opposite one ii) Recalculate the volumetric porosity coefficient The ACELAN modeling of porous medium (e. g., porous ceramics) is carried out at finite-element level. Mechanical properties (i.e., density and elastic modules) and piezoelectric properties (i.e., piezoconstants and dielectric permittivity) of some randomlyselected elements are changed (by multiplication on a coefficient or by exchange on coefficients previously defined) at during assembly of the mass, damping, and stiffness matrix. However, users must take into account two additional circumstances to obtain satisfactory results of modeling. The first is the choice of material damping coefficient by

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defining pore parameters. The second is the size of the finite-element mesh. Its The shape of the mesh should not influence the piezoelement characteristics at the given porosity. The utility and the solver make it possible to carry out a series of experiments, which allow us to draw conclusions regarding the set of parameters of finite-element modeling of composites with irregular pore structure. For this, into framework of the linear theory of electroelasticity [8], the self-oscillations of rectangular bodies in two-dimensional positioning ( ui are the components of displacement vector,  is the electric potential) were considered. One-dimensional piston movements parallel to the vector of preliminary polarization of the square (the side size l  0.009 m) were investigated (thickness modes). Value-boundary conditions of the problem are:

u1  0 , 13  0 , D3  0 at the x  l / 2 ,

 33  0 ,  13  0 ,   0

at the z  l / 2 .

(13)

We chose PZTB-3 as an example of a solid ceramic. The properties of the pore material modeling are defined with the help of reduction coefficient a p  a 10  n , where a p

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denotes elastic modules and the density of porous material, a , is a corresponding mechanical property of solid ceramics. The first series of experiments was directed to detect the optimal reduction coefficient. By using the dependence diagram F of relative frequencies of the first thickness mode fp (F  , f p is the frequency of the first thickness mode of a porous body, f s is the fs frequency of the first thickness mode of a solid body) and the degree of reduction coefficient 4 n (Figure 5), it is possible to conclude that the optimal reduction coefficient is 10 . Further experiments were directed to detect the optimal body discretization. For these experiments, we used the dependence diagram of relative frequencies of the first thickness mode and the number of nodes in the finite-element mesh (Figure 6). On this basis, it is possible to conclude that the number of nodes must be more than 3,300. Numeric evaluation showed that pore re-distribution (volumetric content is 8.66%) did not influence the first thickness mode of 202 kHz (Figures 7, 8).

Figure 5. Relative frequencies of the first thickness mode and the degree of reduction coefficient n .

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Figure 6. Relative frequencies of the first thickness mode and the number of knots in the finite-element mesh

Figure 7. Initial distribution and re-distribution

The components of the stress-strain state of the piezoelement (e. g., displacements of points of the piezoelement surface) may serve as an indicator of pore distribution adequacy. Therefore, the first stage includes a series of experiments aimed at detection of the optimal reduction coefficient. By using the dependence U of relative horizontal displacement (F



up us

, up

is the horizontal displacement of porous body, u s is the horizontal

displacement of solid body) on the degree of reduction coefficient n (Figure 9), it is possible 5

to conclude that the optimal reduction coefficient is 10 . Further experiments were aimed at definition of the optimal body discretization. By using the dependence of the relative horizontal displacement on the number of finite-element mesh nodes N (Figure 10), it may be concluded that the number of mesh nodes must be more than 3,750.

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Figure 8. The first thickness mode of pores in the 8.66%-composite.

Figure 9. Relative horizontal dislocation and the degree of reduction coefficient n.

Figure 10. Relative horizontal displacement on the number of finite-element mesh nodes N.

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4. ESTIMATION OF EFFECTIVE MODULES

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The solver developed was used to define the effective elastic and piezoelectric properties of composite piezoceramics with irregular structure. The piezocomposites are used to construct piezoelectric transducers whose characteristics are determined by the structure of the piezocomposite used. One type of piezocomposites used is porous ceramics with regular

or irregular pore structure [24]. We assume that one possible method of describing piezocomposites is by their replacement by homogeneous compositions having certain effective characteristics [26]. Numerous studies deal with estimating the full set of piezoceramic material constants [25]. Although the many methods exist for estimating piezoceramic material constants [27, 28], we decided to develop an independent estimate of these properties and their operating parameter dependencies (temperature, oscillation frequency, etc.) . Our method for estimating effective elastic and piezoelectric properties of composite ceramics with irregular structure is based on dynamic equivalence of the composite to the homogeneous piezoceramics. We also considered self-oscillation of some representative volumes of the composite and define their resonance and anti-resonance frequencies. All of these calculations are carried out with the help of finite-element ACELAN software. The frequencies for analytically depicting the set of one- and two-dimensional movements provide the basic data for estimating the effective constants. It is assumed that a range of characteristics (dielectric permittivity, density) can be measured according to well-known experimental methods designed for homogeneous materials; they are calculated in computer simulation by using the theory of mixtures. The following oscillation modes were used: a) One-dimensional piston movements parallel to preliminary polarization vector (thickness modes), b) Oscillations perpendicular to the vector of the long bodies with free boundaries and the piston movements in this direction (longitudinal modes), c) Shear oscillations in the direction of the polarization vector. Self-oscillations of rectangular bodies with one-dimensional positioning are considered in a framework of linear electric elasticity theory [8] ( u i are the components of displacement vector and  , is the electric potential. The piezoceramic PZTB-3 was chosen as an example of a solid ceramic. One has the 10

2

following mechanical properties ( C  10 N/m is the elastic module matrix, E is the piezomodule matrix,  10  12 F/m is the dielectric permittivity matrix), namely

0 0 0   7.5 0.5 8   0 0 0   0.5 7.5 8  8 8 13.6 0 0 0    C 0 0 2.9 0 0   0  0 0 0 0 2.9 0    0 0 0 0 0 3.6  

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A. N. Soloviev and G. D. Vernigora

0 0 0 15.4 0   0   E  0 0 0 15.4 0 0    7.9  7.9 17.7 0 0 0  



0 0  14288.5    0 14288.5 0   0 0 11328  

The solution of the problem with boundary conditions (13) we present as

u1 ( x, z )  0 , u3 ( x, z )  U z ( z ) ,  ( x, z )  Ф( z ) .

(14)

The equations estimating resonance/anti-resonance frequencies has the forms

 1 1 l l tg  r t 2  1    r  2 k3  2 k3

k t 2 1 , a  n 3 1  t 2 , 2 t l

(15)

2

e33 cii , i  1,3,5 , r , a are the natural frequencies of where, t  , ki    33c33 2

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resonance and anti-resonance in the thickness oscillations,

ij , eij , cij

are the effective

components of tensors of the dielectric permittivity, piezoconstatns and elastic constants. The boundary conditions of problem B1 are

 11  0 , 13  0 , D1  0

at the

x  l / 2 ,

 33  0 ,  13  0 ,   0

at the

z  h / 2 .

Assuming that l >> h and

(16)

 33 ( x, z)  0 , we present the solution as

u1( x, z)  U x ( x) , u3 ( x, z)  U z ( z) ,  ( x, z )  Ф ( z ) .

(17)

The equations estimating the resonance/anti-resonance frequencies have the following forms

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Identification of Effective Properties of the Piezocomposites…

k1 1 g 2 , l 2 2 1 l 2  2 (1  t )(1  g )c  1  g   a 1 g , k3 t 2 (ec  1) 2  2

235

r  (2n  1) 1 l tg  a k3 2

(18)

e31 c33 c132 , e , c . where, g  e33 c11 c11c33 2

The boundary conditions of problem B2 are

 11  0 ,  13  0 , D1  0 u3  0 ,  13  0 ,   0

at the

at the

x  l / 2 ,

z  h / 2 .

(19)

We present the solution as

u1( x, z)  U x ( x) , u3 ( x, z )  0 ,  ( x, z )  0 .

(20)

The equation estimating resonance frequencies has the form

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r  (2n  1)

k1 l

(21)

The boundary conditions of problem C are

 11  0 ,  13  0 ,   0

at the

 33  0 ,  13  0 , D3  0

x  l / 2 ,

at the

z  h / 2 .

(22)

We present the solution as

u1 ( x, z )  0 , u3 ( x, z )  U z ( x) ,  ( x, z )  Ф ( x) .

(23)

The equations estimating resonance/anti-resonance frequencies have the forms Z  Z  0,

(24)

where

2 g15

1 1 l d2 l  Z  a  tg  a  , d  T . 2 k5 1  d k5  2 s55 11 2

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236

A. N. Soloviev and G. D. Vernigora Thus, the system of Equations (15), (18), (21), (24) at each harmonic presents a separate

system of seven equations, with seven relative unknowns c11, c13 , c33 , c55 , e31, e33 , e15 as effective constants of the two-dimensional problem of electric elasticity theory for 6 mm class materials. It is assumed that dielectric permittivity is found during the experiment. Similar axisymmetric representative volume problems also can be considered to estimate the permittivity numerically and analytically. As a result of numerical experiment, we were able to estimate the elasticity module and piezoconstant

e33

c33

on the basis of Equations (15) and the resonance/anti-resonance

frequencies for representative volume of 17.6%- and 23.0%-porous ceramics with irregular structures using ACELAN software. The results of numerical experiments are presented in Table 3. Table 3. The results of numerical experiments

Piezoceramic constants of PZTB-3

Effective constants of porous ceramics Volumetric content of pores,

Volumetric content of pores,

10%, f r =202.2 kHz,

23.0%, f r =165.5 кГц,

f a =220.4 kHz

f a =184.8 kHz

e33 , C/m2

17 .7

13.717

11.177

c33 , N/m2

13.6  1010

8.131  1010

4.713  1010

We then considered nine static problems for square bodies (with side size l  0.1 mm ).

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The boundary conditions of problem A are found as

u n  0 ,  n  0 , Dn  0

at the

 n  0 ,   0 ,  n  0

at the

un  0 ,  n  0 ,   0

at the

x  0,

z l, z  0.

(25)

We present the solution of problem A as

u1( x, z)  U x ( x) , u3 ( x, z)  U z ( z) ,  ( x, z )  Ф ( z ) .

(26)

The equations estimating vertical/horizontal displacements have the forms

u1   0

c33e31  c13e33 c132  c33c11

,

v1   0

c11e33  e31c13 c132  c33c11

The boundary conditions of problem B are found as

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.

(27)

Identification of Effective Properties of the Piezocomposites…

237

u n  0 ,  n  0 , Dn  0 , at the x  0 ,

 n  0 ,   0, n  0

at the

z l,

un  0 ,  n  0 ,  0

at the

z  0.

(28)

We present the solution of problem B in the form of Equation (26). The equations estimating vertical/horizontal displacements have the following forms

u2  0

c11l c13l , v2   0 2 c13  c33c11 c  c33c11 2 13

.

(29)

The boundary conditions of problem C are found as

u n  0 ,  n  0 , Dn  0

 n  0 , q  0, n  0 un  0 ,  n  0 ,  0

at the x  0 ,

at the

z l,

at the z  0 .

(30)

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We present the solution of problem C in the form of Equations (26). The equations estimating vertical/horizontal displacements and potential have the forms

u3   0

v3   0

3  0

 33c13  e33e31 l

2 2 c   2 e33e31c13  e33 c11   33c11c33  c33 e31 2 13 33



2 33 11 31 2 33 31 13 33 11

‘



c e l 2 , c   2 e e c  e c   33c11c33  c33 e31 2 13 33

c13e31   33c11 l

2 2 c   2 e33e31c13  e33 c11   33c11c33  c33 e31 2 13 33

The boundary conditions of problem D are found as

u n  0 ,  n  0 , Dn  0

at the

x  0,

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(31)

.

238

A. N. Soloviev and G. D. Vernigora

 n  0 ,   0 ,  n  0

at the

z l,

u n  0 ,  n  0 ,   0 at the z  0 , u n  0 ,  n  0 , Dn  0 at the x  l .

(32)

We present the solution of problem D in the form

u1 ( x, z )  0 , u3 ( x, z)  U z ( z) ,  ( x, z )  Ф ( z ) .

(33)

The equation estimating vertical/horizontal displacement has the form

v4    0

e33 c33

.

(34)

The boundary conditions of problem E are found as

u n  0 ,  n  0 , Dn  0 at the x  0 ,

 n  0 ,   0, n  0

at the

z l,

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u n  0 ,  n  0 ,   0 at the z  0 ,

u n  0 ,  n  0 , Dn  0 at the x  l .

(35)

We present the solution of problem E in the form of Equations (33). The equation estimating vertical displacement and potential has the form

v5   0

l c33

.

(36)

The boundary conditions of problem F are found as

u n  0 ,  n  0 , Dn  0

at the

x  0,

 n  0 , q  0, n  0

at the

z l,

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un  0 ,  n  0 ,   0

at the

u n  0 ,  n  0 , Dn  0

239

z  0,

at the

x  l.

(37)

We present the solution of problem F in the form of Equation (33). The equations estimating vertical/horizontal displacement and potential have the forms

v6   0

e33l  33l , 6  0 2 e33  c33 33 e  c33 33 2 33

.

(38)

The boundary conditions of problem G are found as

u n  0 ,  n  0 , Dn  0 u n  0 ,  n  0 , Dn  0

at the at the

x  0, x  l.

(39)

We present the solution of problem G in the following form

u1 ( x, z )  0 , u3 ( x, z)  U z ( x) ,  ( x, z )  Ф ( x) .

(40)

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The equation estimating vertical displacement has the following form

v7 0

e15 c44

(41)

The boundary conditions of problem H are found as

u n  0 , u  0 ,   0

at the

x  0,

u n  0 ,  n  0 ,    0 at the z  l ,

 n  0 ,  n  0 ,    0

at the

z  0,

u n  0 ,   0 ,    0 at the x  l . We present the solution of problem H in the following form:

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(42)

240

A. N. Soloviev and G. D. Vernigora

u1 ( x, z )  0 , u3 ( x, z)  U z ( x) ,  ( x, z )  0 .

(43)

The equation estimating vertical displacement has the following form:

v8  

a c44

(44)

The boundary conditions of problem I are found as

u n  0 ,  n  0 , Dn  0 n  0, n  0 , q  0

at the

at the

x  0,

x  l.

(45)

We present the solution of problem I in the form

u1 ( x, z )  0 , u3 ( x, z)  U z ( x) ,  ( x, z )  Ф ( x) .

(46)

The equations estimating vertical displacement and potential has the following forms:

v9 0

ae15 a11    , 9 0 2 2  c4411 e15 e15  c4411

.

(47)

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We then found an effective elasticity module c12 through the effective elastic module

c11 , defined in the corresponding steady-state problem in two-dimensional statement, and through the elastic module stress statement, as

c11ps. 

c11ps , defined in the corresponding steady-state problem in plane-

2 2 с11  с12 с11

(48)

The elastic module c66 was defined as

c66 

c11  c12 2

(49)

The results of the numerical experiment are given in Tables 4  6. These tables contain the calculation data about full set of effective constants obtained on the base of Equations

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Identification of Effective Properties of the Piezocomposites… (27),

(29),

(31),

241

(34),

0 100, 0  100,

(36), (38), (41), (44) , (47), (48) and (49) with  100 , by using ACELAN software for calculation of

vertical/horizontal displacements and potential at the representative volume of 10.0%-porous ceramic with irregular structure. Table 4. Effective elastic modules of porous ceramics Volumetric pore content, 10% c11, N/m2

c13, N/m2 3.57  1010

c33, N/m2

8.35  1010

c12, N/m2 2.53  1010

7.51  1010

с44, N/m2 2.23  1010

с66, N/m2 2.91  1010

8.15  1010



3.58  1010







Table 5. Effective piezoconstants of porous ceramics Volumetric pore content, 10% e15, C/m2 10.45

E31, C/m2  3.17

e33, C/m2 12.98

Table 6. Effective dielectric permeability of porous ceramics Volumetric pore content, 10% ε11, F/m2

ε33, F/m2

1223.37  10 -12

9752.41  10 -12 9700.43  10 -12

1219.64  10 -12

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ACKNOWLEDGMENTS The work was carried out with partial financial support from Russian Foundation for Basic Research (project codes 05-01-00690, 05-01-00734, 06-01-08041, 07-08-13589, 07-0812193).

REFERENCES [1] [2] [3] [4] [5]

Streng, G.; Fix, J. The Theory of Finite-Element Method; Мir: Moscow, 1977; pp 1  349 Zinkevich, O.; Morgan, K. Finite Elements and Approximation; Мir: Moscow, 1986; pp 1  318 Belokon, A. V.; et al. In Fundamental Problems of Piezoelectic Instrument Making; Tver State University Press: Tver, 2002; pp 171-179. Belokon, A. V.; Bondarev, P. M. Compos. Mech. Des. 2002, vol. 8, 291-308. Glozman, I. A. Piezoceramics; Energiya: Moscow, 1972; pp 1  288

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242 [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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[21] [22] [23] [24] [25] [26] [27] [28]

A. N. Soloviev and G. D. Vernigora Grinchenko, V. T.; Ulitko, A. F.; Shulga, N. A. Electric Elasticity; Mechanics of Coupled Fields in Construction Elements, 5; Naukova Dumka: Kiev, 1989; pp 1  384 Mason, W. Piezoelectric Crystals and their Application in Ultraacoustics; Foreign Literature: Moscow, 1952; pp 1  423 Parton, V. Z.; Kudryavtsev, B. A. Electromagnetoelasticity of Piezoelectrics and Electrically Conductive Solids; Nauka: Moscow, 1988; pp 1  472 Jaffe, B.; Coock, W. R.; Jaffe, G. Piezoelectric Ceramics; Academic Press: London, New York, 1971; pp 1 – 288 Roncari, E.; Galassi, C.; Craciun, F.; et al. J. Europ. Ceram. Soc. 2001, vol. 21, pp 409 – 417 Galassi, C. J. Europ. Ceram. Soc. 2001, vol. 26, pp 2951 – 2958 Zeng, T.; Dong, X.-L.; Chen, S.-T.; Yang, H. J. Ceram. Int. 2007, vol. 33, pp 395 – 399 13. Kumar, B. P.; Kumar, H. H.; Kharat, D. K. J. Mater. Sci. Eng. B. 2006, vol. 127, pp 130 – 133 14. Zeng, T.; Dong, X.-L.; Mao, C.-L.; et al. J. Europ. Ceram. Soc. 2007, vol. 27, pp 2025 – 2029 Zhang, H. L.; Li, J.-F.; Zhang, B.-P. Acta Mater. 2007, vol. 55, pp 171 – 181 Piazza, D.; Capiani, C.; Galassi, C. J. Europ. Ceram. Soc. 2005, vol. 25, pp 3075 – 3078 Zeng, T.; Dong, X.-L.; Mao, C.-L.; et al. J. Mater. Sci. Eng. B. 2006, vol. 135, pp 50 – 54 Zeng, T.; Dong, X.-L.; Chen, H.; Wang, Y.-L. J. Mater. Sci. Eng. B. 2006, vol. 131, pp 181 –185 Chen, H.; Dong, X.-L.; Zeng, T.; et al. J. Ceram. Int. 2006, vol. 34, pp 1 – 6 Nasedkin, A. V. In Fundamental Problems of Piezoelectic Instrument Making; Moscow, 2000; pp 154 – 158 Belokon, A. V.; Nasedkin, A. V.; Soloviev, A. N. In Theor. Appl. Mech.; Donetsk National University Press: Kharkov, 2001; No 33, pp 45-51. Nasedkin, A. V. High School Proceedings. Northern Caucasus Region. Natural Science. 1999, No 1, pp 48-51 Troelsen, A. C # and .NET platform. Programmer Library; Piter: Sankt-Petersburg, 2002; pp 1 – 800 Lupejko, T. G.; Lopatin S. S. Inorg. Mater. 1991, vol. 27, pp 1087 – 1098 Akopyan, V. A. Theoretic and Experimental Estimation Methods of Full Set of Compatible Material Constants in the Electric Elasticity Theory. PhD Dissertation; Don State Technical University Press: Rostov-on-Don, 2005; pp 1 – 206 Getman, I. P.; Molkov, V. A. Sov. Appl, Math. Mech. 1992, vol. 35, pp 501-509 Bondarev P.M. Development of Numerical-Analytical Calculation Methods for Effective Characteristics of Piezocomposites with 3-0 and 1-3 Connectivity. PhD Dissertation; Rostov State University Press: Rostov-on-Don, 2002; pp 1 – 127 Nasedkin, A. V. In Modern Problems of Mechanics of the Continuous Media; Belokon, A. V.; Ed.; New Book: Rostov-on-Don, 2003; Vol. 1, pp 111 – 115

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Chapter 6

TOUGHENING MECHANISMS AND FRACTURE RESISTANCE OF FERROELECTRIC MATERIALS I. A. Parinov1 Vorovich Mechanics and Applied Mathematics Research Institute, Southern Federal University, 2 Rostov-on-Don, Russia

ABSTRACT

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Various toughening mechanisms associated with microstructure features of ferroelectric materials can lead to crack shielding, thus significantly increasing the effective fracture toughness and other strength properties in these materials. In other cases, the crack interaction with microstructure leads to the crack amplification. For this reason, it is important to study characteristic fracture mechanisms for ferroelectrics, define their main features, and couple the technological and microstructure peculiarities with properties of the prepared samples. This study first presents an overview of some methods and models of toughening mechanisms and fracture resistance related to ferroelectric materials. The discussion relates to considerations of microcracking and twinning zones near macrocracks, phase transformations, crack deflection, branching and bridging, and other phenomena. Then, a catastrophic crack growth condition is examined and the fracture energy changes are estimated in dependence on the presspowder initial porosity, and intergranular cracking and phase transformations in vicinity of the crack tip. A computer simulation is developed taking into account the actual physical model of gradient sintering for lead-zirconate-titanate (PZT) ceramics. We also present a fracture model and the results of its numerical realization for PZT ceramics, taking into account hysteretic ferroelectric domain-switching processes near the macrocrack tip. Both models are based on the balance of energy caused by driving forces and on a consideration of the total energy connected with the processes of dissipative interactions around the crack.

1 2

e-mail: [email protected]. 200/1, Stachki Ave, 344090, Rostov-on-Don, Russia.

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1. INTRODUCTION The invention of new ferroelectric materials requires investigation of the formation and degradation of their material strength and the causes of variation in their fracture resistance related to their fracture toughness and strain-energy release rate. Microcracking, phase transformation and domain reorientation taking place near macrocracks in the ferroelectric material are the main cause of differences in these parameters. Computer simulation is a proven method of evaluating these properties and takes into account the preparation requirements and tendency to fracture of various piezoceramic and other related materials [58, 84, 88  91, 93, 97  99]. As a rule, optimization of the strength of these materials requires a consistent treatment of related processes, namely: (i) modeling of material microstructures during processing, (ii) spontaneous microcracking during cooling process, (iii) macrocrack interactions with microdefects and microdamage, (iv) evaluation of influence of toughening mechanisms and crack amplification caused by microstructure transformations on the strength parameters described, (v) determination of the most prevalent effects defining fracture resistance, and (vi) an estimate of total fracture resistance on the basis of force (i.e., via fracture toughness) and energy (via the release rate of strain energy) approaches. This study presents methods and results of investigations of toughening mechanisms and fracture resistance in ferroelectric materials. The discussion focuses on consideration of microcracking and twinning zones near macrocracks, phase transformations, crack deflection, crack branching and bridging, and other similar phenomena. Section 2 presents an overview of theoretical results and model investigations for estimating crack shielding and crack amplification mechanisms. Section 3 discusses model presentations of computer simulation methods. The ferroelectric ceramic (FC) sintering and microcracking during cooling and poling are treated on the basis of the lead-zirconate-titanate (PZT) family. Section 4 discusses catastrophic crack growth conditions and estimates fracture energy changes related to the initial porosity of presspowder, intergranular cracking and phase transformations. The computer simulations include the process of microstructure formation during sintering, spontaneous cracking by cooling, and energy absorption near propagated macrocracks due to phase transformations. Section 5 presents a fracture model for PZT ceramics taking into account hysteretic domain-switching processes near the macrocrack tip. The model is based on the balance of energy caused by driving forces and on a consideration of total energies connected with the dissipation due to ferroelectric domain-switching processes of energy stored in the crack wake zone and energy consumption in the formation of new fracture surfaces. Section 6 offers some conclusions regarding the model investigations and proposes an approach for estimating fracture resistance in ceramics and composites.

2. TOUGHENING MECHANISMS OF CERAMICS AND COMPOSITES In this section, we present an overview of fracture features of ceramics and composites, and review proper toughening mechanisms of interest for strengthening the brittle oxide structure of ferroelectrics.

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2.1. Bending of Crack Front

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The deformation of the frontal zone of a crack occurs as a result of the resistance of secondary phase inclusions meeting the path of the growing crack. These interactions lead to a lengthening of the crack front and its bending between the particles (see Figure 1). By achieving a particular curve, an uneven break occurs at the crack front and then it straightens outside the inclusions. According to the model [65], the energy needs both for as the creation of a new fracture surface as for the distortion of the crack front or its bending and lengthening between the heterogeneities. Before the crack jump (i.e., in the moment of the separation of the crack front from the inclusions), the accumulated energy is stored in the space between particles. Moreover, the fracture resistance of brittle material increases with a decrease in the distance between particles. Thus, the fracture energy and the strength of brittle materials could be increased by the introduction of nearly disposed inclusions, which are capable of slowing down the crack front propagation. It is known that the distance between particles is sometimes less than the actual size of the crack. For this reason, an understanding of the effect of inclusions on the increase of fracture energy is essential. An increase in toughness together with volume concentration of stiff inclusions is supported by the model results [25]. However, the effective influence of inclusions on the fracture toughness depends also on the mutual orientation of the stiff inclusions and the crack front. The heterogeneity form has most value in definition of the front bending. In the case of inclusions in the form of bar or disk, a higher ratio of characteristic sizes h/R (where h and R are the respectively the inclusion height and radius) increases front bending and the corresponding growth of the fracture toughness. This morphological effect diminishes significantly for compliant inclusions, and the bending effect is generally defined by degree of the inclusion permeability [25]. The last effect is characterized by a ratio of toughness of the matrix and inclusions, and also by the strength and coherence their interfaces.

Figure 1. Lengthening and bending of front of a growing crack between particles.

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Figure 2. Typical crack deflections: (a) crack tilt and (b) crack twisting at the interaction of the crack with inclusion.

2.2. Change of Crack Trajectory Material anisotropy is one source of crack trajectory lengthening and of its change in orientation relative to the direction of the action of maximal tension stresses. The crack propagates either under influence of internal residual stresses of II type of the grain scale or due to the existence of weakened interfaces. The former arise from deformation or temperature mismatches in the material microstructure. With weakened interfaces, a dispersion phase with a greater thermal expansion factor or more elastic modules than the matrix causes a deformation of tangential compression near the interfaces and leads to deflection of the crack path around the affected particles. In the case of a lower thermal expansion factor for the inclusion as compared with the matrix, tangential tension occurs, leading to a change in the crack path toward the affected particle [102]. In this way, the sign and value of residual strain insignificantly influence the toughening value caused by interaction of the crack with the inclusion. By achieving the structural heterogeneity, the crack could deflect from its initial plane of propagation. The initial tilt angle depends on particle orientation and disposition into plane of

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the crack propagation. The subsequent growth of the crack may be accompanied by anterior twisting [25] according to the orientation of adjacent particles (see Figure 2). The crack tilt is characterized by I and II Modes of fracture (respectively, due to normal breaking and longitudinal shear). At the same time, crack twisting found by I and III Modes of fracture are due to normal breaking and transverse shear, respectively. Crack deviation from its initial path increases the material fracture toughness and decreases stress intensity near the crack tip. The calculation of toughening caused by changes in crack path may be reduced to solution of a statistical problem involving the definition of the motive force averaged on each possible angle of deflection of the crack segments at the concretization of microstructure features of the sample considered. Obviously, the most tortuous fracture trajectory and the increase in the number of separate crack segments lead to an increase in the toughening value. Thus, the grain shape with greater characteristic ratio influences greater change of the motive force of the crack. Moreover, an increase in particle concentration promoting change in crack orientation also increases material toughening (see Figure 3).

Figure 3. Increase in release rate of strain energy in dependence on volume fraction of the particles of various forms rounded by crack [15].

Note in Figure 3 that the most effective morphology of inclusion deviation of crack orientation is for the bar, with a high characteristic ratio leading to a fourfold increase in fracture toughness [25]. Lesser effects of toughening are shown in particles in the shape of discs and spheres. Fracture toughness does not take into account the inclusion morphology in greater degree is defined by the crack twisting but no its initial tilt. Only in the case of disc

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inclusions does the initial tilt of the crack front lead to significant toughening of material. However, in this case, crack twisting also defines higher values of toughening. For all three morphologies (bars, discs, and sphere), we observe a weak increase of toughening at a concentration of dispersed particles greater than 0.2. Moreover, in the case of spherical inclusions, most of the toughening is achieved at the maximal approach of inclusions one to another. Crack propagation is suppressed when the twisting angles achieve values near 90º. Thus, an ideal fraction of secondary phase inclusions are at values of 10  20 % [25]. Experimental data obtained for different materials confirm these results [26].

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2.3. Microcracking During Ceramic Processing As a rule, spontaneous formation of microcracks at intergranular boundaries, initiates during ceramic processing as a result of sample cooling after sintering and also under influence of thermo-mechanical loading and electro-magnetic fields. With material cooling, microcracking is caused by anisotropy of the thermal expansion factor (TEF) of grains, deformation mismatch, and the like. As noted in the analysis of stress-strain state of singlephase polycrystal ceramics, the initiation of microcracks depends on microstructure parameters, namely size, form and mutual orientation of adjacent grains, external loading and physico-mechanical material parameters. One of the first problems investigated is the problem of anisotropic grains surrounded by stiff matrix with mean value of TEF of the polycrystal [67]. Many known models indicate a deterministic character: for example, that of a polycrystal consisting of a certain number of two-dimensional hexagonal grains in an infinite matrix and possessing elastic isotropy and anisotropy of TEF of the adjacent grains with arbitrary 3 orientations. This model uses Eshelby‘s conceived procedure as a solution method [20] and is very convenient due to its natural consideration of triple points. As shown in tests [21], the last sites localized the microstructure defects for example voids, secondary phase inclusions, admixtures. Thus, they may be considered as the nuclei of microcracks, which form at grain boundaries. Theoretical analysis [57] shows stresses demonstrating logarithmic singularities near triple point for TEF anisotropy used for estimating the critical size of faceted microcracks. References [21, 57] investigate the influence of the relative size of the defect into triple point 2a/r (where 2a is the defect size and r is the length of grain boundary) on the critical size. It has been noted that the critical grain size increases with the decrease in characteristic size of triple point for various values of fracture energy of the grain boundary. The investigation of different fracture Modes presented in Reference [35] is dependent on the triple point size. For regular orientation of adjacent grains, the numerical results show that the stress intensity factor KII is significantly lesser than KI. Thus, in the cases considered, microcracking is essentially found by Mode I. The formation of a lattice of microcracks leads to a decrease in elastic modules caused by a volume fraction of microcracks. External stresses may lead to the formation of intergranular cracks that are shorter than those generated by critical stresses for spontaneous cracking and 3

Figure 4 illustrates approximate scheme that investigation [34].

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increased density of damage. Reference [66] investigates the dependence of that threshold stress on the size of the cracked boundary. That study also presents microcracking criteria that connect facet size with critical one and include misorientation of adjacent grains in twodimensional cases. Reference [83] investigates the influence of elastic anisotropy on ceramic cracking by finite-element method. That study is especially actual one for materials of cubic symmetry in the absence of TEF anisotropy. In this case it appears that in the vicinity of triple point, stresses demonstrate singularities of power type. While the whole elastic effect expected is not great compared with anisotropy of thermal expansion one, the local fluctuation of stresses should exert certain effects on strength properties of the sample. Generally, the stress-strain state is caused by temperature change, degree of the TEF anisotropy and elastic modules. As stated, the influence of elastic anisotropy compared with that of the TEF anisotropy in small deviation from homogeneity to be effect of second order and in first approximation one could be neglected.

Figure 4. Scheme demonstrating Eshelby‘s method for calculating internal stresses and deformations caused by TEF anisotropy

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The problem of subsequent cracking of matrix surrounding a spherical grain with a nucleus equatorial crack is investigated in Reference [62]. This study shows that spontaneous cracking is found by the grain size, elastic properties, material fracture toughness and length of the nucleus crack.

2.4. Microcracking in the Stress Field of a Macrocrack

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The toughening mechanism of a material, caused by microcracking into a microcracking zone of a macrocrack, is connected with the process of microcrack initiation at intergranular boundaries under critical stresses near a macrocrack. This phenomenon leads to crack shielding and demands higher stresses for propagation of the next crack. In this case, the ceramic microstructure promotes the release of elastic energy from internal stresses formed during material processing. The shielding effect in the crack tip is schematically shown in Figure 5. In the absence of a microcracking process zone around the crack, the stress into the crack plain is proportional to applied stress intensity factor Ка and decreases with distance r measured from the crack tip according to law 1/r1/2. The crack shielding diminishes stress around the crack tip, due to alteration of the stress function to the lower dashed line in Figure 5. Thus, beyond the process zone, the stress field is induced by applied stress, at the same time these stresses decrease owing to the crack shielding.

Figure 5. Stress state near crack tip in absence of process zone (upper dotted line) and in existing crack shielding zone (lower dashed line), where K0 and Ка are the motive force of the crack propagation and applied stress intensity factor, respectively.

Changes in ceramic fracture toughness are typically studied using two approaches: (i) by investigation of ‗spreading‘ microcracking effects with corresponding decrease in elastic modules into process zone, to define the zone of abundant microcracking ; (ii) by studying the

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interaction of the macrocrack tip with discrete microcracks. Analogously, in the case of plastic material, crack blunting is explained either by using a presentation of plastic zone into framework of mechanics of continuous media, or by examining a micromechanical model based on dislocation emissions and interaction of the dislocations with the macrocrack [23]. Crack interactions with discrete microdefects is discussed in the next section. Here, we consider the formation of the microcracking process zone and its influence on fracture resistance. The effect of decrease in elastic modules may be estimated either by considering the influence of surrounding materials on the process zone, or by investigating constitutive relationships in process zone. In the first case, toughening is caused by the influence of stiffer surrounding material on the softer defect process zone. In the second case, microcracking leads to a change in dependence of stress on strain    (see Figure 6) near the crack tip [23]. Changes in fracture toughness reflect opposite trends caused, on the one hand by a compliance of the microcracking zone, and on the other hand, by a decrease in local fracture toughness due to microcracking immediately adjacent to the macrocrack and defining the processes of crack coalescence (see Figure 7). The phenomenon of microcracks increasing fracture toughness assumes that a discrete zone of microcracking develops near the macrocrack, but that microcrack formation into the zone immediately surrounding the macrocrack is suppressed. Taking into account the TEF anisotropy of the grains, Reference [34] presents a comparative study of preference formation of discrete cracks compared with the subsequent propagation of the macrocrack along the boundary of hexagonal grain disposed at the crack tip (see Figure 7).

Figure 6. Non-linear dependence    defining the process zone. In the microcracking saturation zone, the Young‘s module Es is maintained constant and caused by the density of microcracks.

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Figure 7. The formation of crack deflection and secondary cracks (discrete microcracking) at the macrocrack growth into polycrystal.

The use of Eshelby‘s procedure (see Figure 4) and the comparison of corresponding stress intensity factors for both cases show that microcracks are formed before the propagation of the main crack when the grain size of the material is greater than the critical value. The grain size here is significantly less than that of grains defining spontaneous Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

microcracking during cooling l cs . Discrete cracking caused by macrocrack stresses takes place at grain sizes lc equal to  40% of the critical size, corresponding to spontaneous microcracking. It has been stated that discrete cracking increases fracture toughness when grain size corresponds to the condition l > lc. However, for the value l  l cs , spontaneous microcracking again initiates more no leading to increase of the crack shielding. The next raise of the grain sizes decreases fracture toughness due to growth of spontaneous cracking and process of the microcrack coalescence. The following increase of toughness in this range may be only ensured by formation of the grain-bridges pinning the crack faces or branching of micro- and macrocracks. An important characteristic of fracture toughness alteration at the crack growth is the socalled R – or Т – curve (respectively in energetic and force approaches), which defines the fracture resistance of a ceramic. This value characterizes the energy (or toughness) alteration in dependence on the length of growing crack. In general, this curve is not a monotonousincreasing function, as assumed in Reference [108], due to the processes taking place at the crack tip. When the maximal principal stress achieves the threshold value th in monotonousincreasing tension, rapid growth of the stress state begins near the crack tip and leads to an increase in microcrack density. When formation sites for microcrack nucleuses are exhausted,

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the microcracks cease to initiate in grain boundaries and begin to form in the saturation zone immediately adjoining the crack tip (see Figure 8). Within the framework of this assumption, the source of the behavior of the R – curve has been studied [23]. The use of the J-integral method [108] shows that microcracking into saturation zone weakens the resistance to local fracture. However, further computation demonstrates only weak decrease in toughness

K c / K c0  0.9 , where K c and K c0 are the fracture toughness caused by applied loading

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and fracture toughness of the material without defects, respectively. This behavior does not depended on grain size, process zone or constant deformation. Result concludes that the microcracking into crack frontal zone is not the actual toughening mechanism.

Figure 8. Configuration of frontal process zone formed at the crack growth into material which is initially free from microfractures.

The investigation of steady-state crack growth (see Figure 9) by defining the fracture toughness on the base of relationship of the energetic balance shows that maximal value of fracture toughness is achieved at l  l cs [23]. Moreover, the toughness increases with the grain size into range l c  l  lcs and is accompanied by growth of in the width of the saturation zone. Then, the total energy, obtained by integrating the entire width of the process zone, increases also and define the role of the width of the entire zone in changes in ceramic toughness. The growth of the process zone width when grain size is close to the critical value for spontaneous cracking has been examined by finite-element method applied to an array of hexagonal grains [57]. Reference [45] investigates the crack shielding or decrease of stress intensity at the crack tip, surrounded by the microcracking zone, in the cases of growing and stationary cracks. This study shows the contribution in the crack shielding due to a decrease of elastic modules at isotropic distribution of microcracks to be approximately 40 % greater in growing cracks compared with stationary ones. Investigation of different conditions in microcrack initiation

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shows that microcracks perpendicular to maximal principal stress lead to more effective crack shielding at the same level of microcrack density. Finite-element analysis [82] of the stress state of the crack under mixed loading Mode shows that microcracks shield the crack tip less in the case of Mode II as compared with Mode I. Moreover, the asymptotic fields for various mixed Modes show the existence of an elastic wedge region in which the material has not cracked. This phenomenon leads to an absence of shielding of the crack tip from loads, acting in far zone, at the proper wedge angle.

Figure 9. Configuration of process zone formed around a crack during its steady growth.

The critical value of microcracking dividing the zones of toughening and microcrack coalescence has been stated analytically [125]. It is usually assumed that in this case the critical value is equal to 0.5 [23]. However, Reference [125] stated that the value to be in reverse-proportion to the third power of the characteristic size of microcrack. The release rates of deformation energy, caused by the macrocrack propagation into its surrounding defect zone, and also the interaction energy of the crack with the defect zone are analyzed in Reference [14]. This study applies semi-empirical analysis of stresses by means of a division of the defect zone around the crack into two regions: a region of high density and a region of low density. In the region of high density of microcracks, immediately adjoining to the crack tip, the mean density of microcracking in conditions of zero stress intensity is estimated. In the low density region, including peripheral microcracks of low density, the density of microcracking is measured experimentally. The numerical results show that the defect zone can effectively shield the macrocrack and diminish the existing energy at the crack propagation into the defect zone. In this case, the most of the shielding effect is caused by the microcracking zone adjacent to the crack. Moreover, a release of elastic energy

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connected with growth of the defect zone occurs in the absence of macrocrack propagation. In this case, the peripheral microcracking zone most contributes in the macrocrack shielding. Finally, the study shows that the relative energy G1 / G 2 (where G1 is the release rate of deformation energy in whole caused by the defect zone but also defined by the macromicrocrack interactions, and G 2 is the release rate of deformation energy caused by interaction of the crack with defects) increases with crack lengthening.

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2.5. Interaction of Macrocrack with Microdefects A second theoretical approach, connected with investigation of toughening at the ceramic microcracking, concerns the problem of the macro-microcrack interactions studied using models of arbitrarily oriented microcracks near the macrocrack tip. The aim of these investigations is to estimate a change (i.e., increase or decrease) in stress intensity factors (SIFs) at the crack tip according to the size, orientation and disposition of microstructure defects. Standard configurations of systems of interacting cracks are presented in Figure 10. The exact analytic solution for the system presented in Figure 10a is stated using Westergaard‘s complex potential and shows a decrease in fracture toughness at the micromacrocrack interactions studied [110]. Figures 10b and 10d, present solutions obtained by using complex potentials corresponding to presentation of the point source of perturbation [109]. In the particular case ( = 0), when microcracks are parallel to the macrocrack (see Figure 10e), the critical value of angle  defines a transition from a crack shielding effect to a crack amplification effect. The corresponding angle of neutral shielding  = 0 is found by ratio of length of the proper microcrack (2s) to distance from the macrocrack tip (r). It is also shown that in wide bounds of change in the ratio s/r, insignificant changes occur from 0 = 70 (at the s/r  0) down to 0 = 69.4 (at the s/r = 2/3). The values of the angle  < 0 define the region of crack amplification, but at 0 <  <  the zone of crack shielding at the macrocrack tip indicates a maximal effect of crack shielding existing in the wake zone of the crack. Macrocrack interaction with microcrack (see Figure 10b) or circular cavity (inclusion), shown in Figure 10c, was investigated by using complex potentials of stresses into framework of linear elasticity. The problem is reduced to a solution of the I-type singular integral equation on a semi-infinite range [113]. Numerical results show that the critical angle of neutral shielding for a microcrack is approximately equal to 62, and for circular cavity, 75 (at the r = 1.5R). The cavity leads to a higher value of local SIF (compared with the microcrack case) required for coalescence of the macrocrack with the microdefect. Noncollinear cracks surrounding the crack tip (the case of small values ) cause greater growth in local stress intensity compared with collinear cracks (at the same distance between crack tip and microcracks). Moreover, in the case of small misorientation of the macrocrack compared with microcrack ( < 20), there is an effect of free surface, or deflection of the original crack path in the opposite side of the microcrack.

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Figure 10. Two-dimensional configuration models showing: (a) collinear microcrack ahead of macrocrack. (b) single microcrack near macrocrack. (c) circular cavity (or inclusion) near crack. (d) dislocation emission near macrocrack. (e) two microcracks symmetrically-disposed relatively macrocrack.

Reference [17] applies the method of double layer potentials to the loading scheme presented in Figure 10d within a framework of the plane problem of elasticity theory. Analytic solutions have been also found for cracks with finite length interacting with array of N rectilinear microcracks. The numerical results obtained for the case of two microcracks of arbitrary orientation (N = 2) define a change of critical angle of the neutral shielding into range 0 = 55  70. Reference [76] investigates the influence of separate microcracks on SIFs at the macrocrack tip under independent loading of Mode I, II or III and joint loading of

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Mode I and II into framework of the model presented in Figure 10b. By limiting our solution of the corresponding problem to the principal terms of rows, we obtain the next representations for SIFs at the macrocrack tip [76]:

K IMA  K I   A11 (, ) A12 (, ) A13 (, )  K I   MA     s 2    K II   K II      A21 (, ) A22 (, ) A23 (, ) K II  ,   8r   A (, ) A (, ) A (, )    MA    K K 32 33  31  K III  III III    

(1)

where KI, KII, KIII are the applied SIFs in absence of microcracks. The factors Aij (, ) in Equation (1) may be present as: A11(, ) = 2cos (2 + ) + 4cos (2  ) + 8cos (2  2)  6cos (2 3)   8cos (2  4)  3cos (3) + 8cos (2) + 11cos , (2) A12(, ) =  6sin (2 + )  8 sin (2  ) + 6 sin (2  3)  8 sin (2  4) + + 9sin (3)  8 sin (2) + 15 sin , (3) A13(, ) = A31(, ) = 0,

(4)

A21(, ) = 2 sin (2 + )  6 sin (2 3)  8 sin (2  4) + + 3 sin (3)  8 sin (2)  9 sin ,

(5)

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A22(, ) = 6cos (2 + )  4cos (2  ) + 8cos (2  2)  6cos (2  3) + + 8cos (2  4) + 9cos (3)  8cos (2) + 15cos , (6) A23(, ) = A32(, ) = 0,

(7)

A33(, ) = 8 cos (  /2)cos (  3/2).

(8)

Based on Equations (1)  (8), the results outline the contours of the zones of the crack shielding and crack amplification in dependence on angles  and . The problems of macro-microcrack interaction also are studied in 3-D statement (see, for example [37]). The methods and obtained solutions are more complicated as compared with those used for corresponding 2-D problems. However, the qualitative results for both cases are analogous.

2.6. Crack Branching A rapidly propagating crack is an unstable system. It is inclined to constant deflection from the main direction of propagation, but shows very low inertness as its velocity increases, and achieves branching regimes that make the crack vulnerable to any influence. In these conditions, it is probable that the breaking of state of the stable dynamic propagation for crack

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and initiation of the branching processes. After each act of branching, the rate of the crack growth decreases sharply (in some cases from 2 km/s down to total stop [30]). The causes of this phenomenon are the rising fracture surface and the alteration of the angle distribution of stresses at tip of the branching crack. In this case, energy decreases into a closed system, that is, an ‗external loading  test sample‘ effect. That effect defines the crack branching process as an effective mechanism of alteration of the toughening and fracture resistance of ceramics. Crack branching has been observed in brittle materials, for example in ceramics of noncubic symmetry at various grain sizes [109]. In this case, first, fracture energy increases with the grain size, but when maximal value is achieved, the energy begins to decrease with the following increase in grain size. The supposition is that the causes of crack branching are connected with the thermal expansion factor anisotropy and interphase deformation mismatches. The analytic model for bifurcation of a crack with a forked tip created by short growth originating from the tip under the same angles is presented in Reference [119]. In the case of semi-infinite cracks under conditions of Mode III loading, the increase in material fracture toughness and fracture resistance caused by the non-regularity of the crack front also are estimated. Fractal models of branching crack have been studied in Reference [80] on the basis of Mandelbrot‘s fractal geometry [72]. A complete analysis of crack branching in micro- and macroscale is presented in Reference [43]. This study shows that in the case of cracks branching on microscales, the fracture toughness increases exponentially together with the branching angle of the microcracks. In macroscale branching, monotonous growth of fracture toughness increases with the crack branching angle, which coincides with data presented in Reference [119]. The investigations of the crack branching in both the absence and the presence of microcracking processes are presented in Reference [36]. Crack branching when one propagates through a microcrack array is accompanied by coalescence processes (see Figure 11). The critical strength parameters are defined by the ratio of the process zone size to the length of proper microcrack (2p/am), microcracking density (m) and crack branching angle (). However, if the mean value of am is estimated in tests during of ceramic processing, then the value of m is more correctly defined by using computer simulation. The simple estimations obtained on the basis of experimental observations and analytic presentations for other two parameters are absent. Thus, the model of crack bifurcation under conditions of Mode III loading [119] using the conform transformation method defines the crack branching angle of 78.48. At the same time, Reference [118] predicts the crack branching angle 15  18 using the criterion of minimum energy density in Mode I dynamic loading conditions. The wide range of change of the crack branching angle (up to 73), obtained using the various criteria of instability at the tip of the dynamic propagating crack taking into account effects of stress-strain state and experiment data, is also published (see Reference [116]). Analogous results are obtained by defining the size of the crack branching zone near the crack tip. The behavior of numerous microcracks (in particular, damage accumulation) preceding the origination point of the main crack was studied in Reference [80]. Based on statistical approach and fractal geometry, the investigation of that process is reduced to a consideration of a self-similar fractal cluster. Start of its avalanche-like growth is interpreted as the start of macrofracturing.

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Figure 11. (a) The model of crack branching and (b) macrocrack tip demonstrating zones of crack branching (2р) and microcracking at the crack tip (2hm).

Reference [4] proposes criteria for estimating the duration of sub-critical crack growth, which could allow one to effectively control crack propagation at the external loading. Based on analysis of the energetic balance for loading of solid with crack, it is considered the crack branching in a 2-D statement. The present model takes into account local stresses at the tips of secondary microcracks but neglects interaction effects and superposition of a stress state in the vicinity of the cracks.

2.7. Crack Bridging The influence of microstructure features of material on decrease of stress intensity in the vicinity of the crack front has been investigated by many researchers, beginning with Barenblatt, Dugdale and Khristianovich, 1950-60. Other similar factors relate to the presence of plastic metal particles in brittle matrix (see the next section for details) and brittle fibers in composites [15, 46]. Crack bridging or toughening of brittle composite by fibers (or grains) displays in the wide ranges of change in characteristic structure sizes. These microscale effects are related to surface and intermolecular forces similar to Van der Waals forces and act only at distances of some angstroms. The toughening mechanism of crack bridging is effective in macroscale at the mechanical dragging of sub-critical cracks. Investigations of brittle monophase polycrystal materials without transformation toughening indicate that grain-bridges are natural microstructure couples remaining intact far behind the propagating crack tip (see Figure 12). This phenomenon exists where the stress concentration is

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exceptionally high in zones of microstructure coupling of crack faces formed due to grainsbridges. The bridging role of grains may be generalized as follows. At short crack lengths, the probability is very small that the grains will pin crack faces and create a bridges. In this case, fracture toughness of the material will be defined by the fracture toughness intrinsic to the intergranular boundary. After the crack propagates, grain-bridges that remains intact will pin subsequent crack growth and impede an opening in its faces. In this case, the grain-bridge is subject to tension. By achieving a certain critical value corresponding to strength of the grain, the transgranular fracture of the bridge occurs and leads to the crack jump until it collides with another grain-bridge. Experimental and theoretical investigations of this toughening mechanism define a change in fracture resistance described by the behavior of the R- (or Т-) curve [6, 7, 123].

Figure 12. Region behind growing crack: (a) before and (b) after the crack propagates. Grain-bridge is clearly observed in the right-hand micrograph [15].

Reference [7] explores the parameters of Т-curve for three types of microstructure defects: (i) a sharp microcrack co-measured with the grain boundary and with a preliminary history of Т-curve development, taking into account deformation of the couple before crack propagation, (ii) a microcrack without preliminary history of a Т-curve or with an absence of 4 previous deformation of the couple and (iii) a relatively large pore. Experimental data for large-grained ceramics of alumina obtained in conditions of external loading created by Vickers indentation states that two parameters of the Т-curve should be maximized in order to achieve an optimal, controlled fracture of ceramic, namely (i) value of Т/T0, where Т, T0 are 4

This case may also include possible fractures of the couple during the material processing due to, for example chemical or/and thermo-mechanical influences.

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the respectively the ceramic toughness at the steady-state of crack and toughness of intergranular boundary, (ii) limit value of c  / d , where c  is the crack length at which it is started a fracture of the couples having characteristic size d and to being most far from the crack tip. Mechanism of grain-bridge formation caused by residual temperature stresses are also taken in to consideration, along with the control of changes in material toughness during uncoupling and pulling out the grains by surfaces of the growing crack [6]. The model allows an estimate of toughness and strength of ceramics depending on size, shape and disposition of grains that drag the crack; and also on grain boundary energy, residual stress states and the sliding friction factor of the grains. Moreover, it shows that it is mainly internal stresses that cause the changes in toughness during the pulling out process as compared with the process of uncoupling in adjacent grains. In addition to unimodal homogeneous and rectangular grain structures addressed in Reference [6], Reference [123] investigates a bridging grain of arbitrary form and spatial orientation. It is assumed that the stress state pinning the crack and fixing the grain-bridges at their starting point are a result of morphological features and residual stresses also caused by TEF anisotropy of the grains. The modeling shows that only small fraction of the grains contribute to the process of crack dragging. Three mechanisms of energy dissipation, caused by the formation of grain-bridges behind the front of the growing crack, were investigated, namely (i) the bending of elastic couples (grain-bridges) at the fracture, (ii) energy dissipation due to opposite action of the wedge grain on the coupling of crack faces, and (iii) twisting of the wedge pinning grains, which influences the surrounding grain structure. These results shows that the most effective and probable mechanism is twisting of the wedge pinning grains. This conclusion is also confirmed by experimental data obtained by methods of electronic microscopy and coincide with the results of Reference [6]. Finally, also observed is an increase in the toughening rate (inclination of the R-curve) together with increased growth in grain size and the dependence of toughness on the grain size defined at the steady-state crack propagation.

2.8. Ceramic Toughening by Plastic Phase Numerous studies describe the toughening of brittle materials by disperse plastic phase (see, for example [2, 81]). Two toughening modes of this type are governed by the plasticity of the disperse phase. The particles crossed by the crack demonstrate intensive plastic elongation into the crack wake zone when strongly coupled with the matrix (see Figure 13), and they contribute to the increased toughness due to the effects of the plastic inclusions bridging the crack [24]. When the particles are near the bridging zone, residual stresses in material due to the difference of TEF of the phases also may contribute to toughening, defining the initial force of the crack opening. Simultaneously, the plastic deformation of inclusions into the process zone leads to crack shielding (see Figure 13). The first mechanism, in general, characterized by high toughness, is found, through great effort, at the particle fracture and in large inclusions. By contrast, the processes of crack shielding demonstrate a tendency to increase in the case of small inclusions with low ductility limits [117]. The

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deformation mode of the plastic phase could be described by investigation of the fracture surface and coupling zone near tip of the surface crack.

Figure 13. (a) Micrograph of couple zone into composite Al2O3/Al, demonstrating elongated Al inclusions between crack surfaces [117], and (b) schematic illustration of crack bridging process caused by crossing of particles and by an existing process zone of plastic deformation at the inclusions.

Detailed models of the bridge formation for brittle composites with plastic inclusions are based on their modeling by distributed non-linear elastic couples presented by springs [9, 111]. An approach presenting the solution in the form of sliding lines in conditions of plane strain [24], was developed to analyze the plastically deforming pinning couples under tension loading and realized by using finite-element method analysis [44]. All of these solutions are based on the assumed formal dependences ‗stress  tension‘ (  u), as a rule, defined into experiments. Fractographic studies of toughened composites and completed experiments describe some distinctive features of behavior of the particles pinning the crack (see Figure 14). In the case of very small stresses, the particles demonstrate almost elastic behavior with an insignificant zone of plasticity near the rounded tip of the crack touching the particle (see Figure 14a). The stress described in this case is similar to that of classic case of elastic-plastic small-scale ductility. Thus, the solution for the sliding lines could be used when the zone of absolute plasticity develops into particles in plane deformation at a remaining high level of

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pinning of the crack faces (see Figure 14b, c). In this case, the stress state into particle is very sensitive to the processes accompanying a blunting of the crack tip and should include relatively high levels of the hydrostatic loading. The constraining conditions of the stiffer surrounding the matrix can create lighter strains at the crack tip and/or evoke delamination processes either interface sliding, allowing a rupture to be formed by pinning couple (see Figure 14d). This process assumes lower levels of stresses into the particle. This situation actually dominates at the final stage of deformation resulting from couple fracture in cases of high plasticity [44].

Figure 14. (a)  (c) Evolution of plastic zone into particle (0 is the ductility limit) and (d) illustration of crack tip blunting used for stress estimation superimposed on the plastic particle (V1 is the removed volume for creation of the pore of radius R, V2 is the additive volume for creation of rupture, the condition of remaining volume is V1 = V2, u and a are the geometrical parameters).

2.9. Transformation Toughening Tetragonal particles ZrO2 demonstrating martensitic (tetragonal  monoclinic) phase transition are used to toughen high-temperature superconducting ferroelastics YBCO, possessing as ferroelectrics with perovskite structure [41]. Therefore, we consider the

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martensitic transformations of ZrO2 in the stressed state of a macrocrack leading to the material toughening [40]. Three features of the tetragonal  monoclinic phase transition in ZrO2 are of interest, namely [15]: (i) the martensitic phase transition occurs in absence of diffusive processes, (ii) the elementary cell increases in volume in 4-5 % at the transition from high-temperature (tetragonal) phase down to low-temperature (monoclinic) phase, and (iii) the temperature of the phase transition , and to a lesser degree, the volumetric change of the material, are found by doping with an appropriate oxide (for example, Y2O3, CaO, MgO or CeO2). The stabilizing oxide additives the martensitic transformation and fracture toughness during the material processing also may be investigated. Models of transformation toughening [22] define differences in fracture toughness as related to the criterion of martensitic transformation (or the shape of the transformation zone) and to the square root of the width of the process zone. In general, the experimentally measured toughness is related to the dilatation transformation into a zone whose shape is determined by a critical value of shear stresses at the crack tip. As noted, one of the interesting features of all the toughening mechanisms caused by the process zone is an increase in toughening with the growth of its own zone and the wake region (at the crack propagation) leading to potential effects of R- (or T-) curve. References [22, 75] present analyses of the development of the dilatation transformation zone at the crack propagation and the influence of the zone on the material toughening. At the same time, the constitutive equations for the release rate of strain energy [found in 10, 22] state that toughening is found only by transformations in the crack wake zone. The shielding influence of the process zone may be estimated using the Eshelby procedure (see Figure 15) [19], which forms the basis of a McMeeking  Evans model developed for estimating transformation toughening [75]. In whole, the stresses in the process zone defined using the Eshelby procedure are represent by a complex function of zone form, inclusion concentration, transformation deformation, and elastic properties of the process zone and surrounding material. After defining stresses in the zone, the McMeeking  Evans model uses the method of weight functions [11] to estimate the forces of the crack closure in the final stage caused by this stress state. In this case, SIF is estimated by integrating of loads acting into the process zone on surface of the zone, but taking into account the weights defined by the distance from the crack tip. In the case of steady-state of the crack, the above approach describes a change in the fracture toughness into process zone and crack shielding due to the martensitic transformation. The increase of fracture toughness can be estimated in other ways. One approach investigates the toughening caused by pure dilatation and defines the exact energy change accompanying the transformation [73]. A second approach involves computation of stress intensity near the crack which is defined by a constitutive equation for the elementary volume subjected to the stress state of the crack [10]. Both analytic methods of determining the increase in fracture toughness are based on hysteretic behavior of dependence    into the process zone defined by the loading of elementary volume due to stress state created by the crack and by the unloading that follows crack propagation.

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Figure 15. Scheme presentation of Eshelby‘s procedure for estimation of stresses constraining crack opening and caused by dilatation transformation. The material parameters: strain at the dilatation transformation (Т), particle concentration (Vt), Young‘s module for transformation zone (Еz) and surrounding matrix (Еm), Poisson‘s ratio ( )

2.10. Effects of Re-Construction of Domain Structure in Ferroelectrics and Ferroelastics As noted, high-temperature superconductors (ferroelastics) as ferroelectric materials possess the perovskite structure which allows comparison of the effects of typical processes of damage and fracture common to both families. References [74, 103, 104] are among the first experimental investigations of strength and fracture toughness of the ferroelectric ceramics. The observed phenomena are explained qualitatively by microstructure features and in particular by internal stresses [31, 32, 55, 68, 69], but the energy dissipation by the processes of ferroelectric domain re-orientations [77, 103, 127]. Reference [121] presents a

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general overview of fracture processes and dielectric breakdown including the effect of the crack shielding and crack amplification . References [18, 28, 29, 33, 56, 78, 79, 114] constitute systematic experimental investigations of the influence of the electrical field on crack growth and the behavior of the R-curve. In particular, the quantitative explanations of test results based on estimation of residual stresses caused by the processes of ferroelectric domain switching are presented in References [28, 114]. At the same time, we still lack a simple definition of the correct criterion of fracture in vicinity of the crack tip because the theoretical and experimental results relating to the observed fracture toughness of ferro-piezoelectrics studied in dependence on applied electric field are contradictory. Study of the saturation zone near the crack tip is of the paramount importance for a satisfactory explanation of the corresponding data. References [8, 38, 39, 59, 112] describe the effects of the non-linear zones. The physical causes of non-linear behavior of materials is connected with the processes of the ferroelectric domain switching (re-orientations of elementary cells of martensitic type), which are typical for ferro-piezoelectrics. Except for the effects of saturation described, the processes are connected with energy dissipation and re-distribution of internal fields near the crack tip. The pointed zones of domain switching are similar to the zones of phase transformations in the vicinity of a crack. As noted above, one of the first proper models of toughening [10] investigates fracture toughness alteration due to pure dilatation transformation. The dilatation and shear transformations [64], reversible transformation into crack wake zone [120] also have been studied. The application of these theoretical conception to ferroelectrics is described in References [128, 126]. However, the analysis of the field around crack with nonlinear zone of the domain switching described in these last studies do not take into account coherence effects in constitutive equations. Moreover, the studies consider only the crack frontal zone; the effects of the wake zone are neglected. At the same time, the last effect, i.e., the re-orientation of domains into the crack wake zone, exerts the most influence on the steady-state of the crack growth and corresponding changes in fracture toughness. Studies of the various toughening mechanisms (including the domain re-orientations in the vicinity of crack) proper to ferro-piezoelectrics and corresponding mathematical models are presented in References [53, 54, 84  87, 92, 95, 96, 100]. As shown in experiments, the high-temperature superconducting ferroelastics of YBCO family possess an ‗effect of shape memory‘. This property assumes a martensitic mechanism of the stress relaxation at the slow stage of material oxidization. Therefore, a decrease of residual type II stresses, initiating in the vicinity of propagating macrocrack, also may occur due to the martensitic mechanism accounting for energetic advantageous of re-construction of the domain structure of crystals. References [3, 101] present investigations of the processes of twinning in ferroelectrics and ferroelastics caused by the structure phase transitions in the vicinity of crack and the arising local stress states. References [90, 94] present studies of influence of the martensitic transformations and re-construction of domain structure on toughening of high-temperature superconductor YBCO.

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3. PZT CERAMIC SINTERING AND MICROCRACKING DURING COOLING AND POLING We shall use the physical model of gradient sintering for PZT ceramics described in Reference [5], The first basic problem we consider is a quasilinear equation of heat conductivity together with FC microstructure formation due to Monte-Carlo procedure of nucleation and growth of the grains in crystallization process from melt [52]. It is assumed that the model of front of sintering with unit thickness is performed as a two-dimensional lattice containing 1,000 cells, with the square cell size  arranged in a square pattern. Each cell represents a grain or pore of an original press-powder compact. This press-powder sample is placed into furnace, and it is assumed that the temperature distribution within the furnace depends on only one coordinate T(x) and consists of a constant and linear temperature

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curve. The initial porosity C p0 of the original press-powder compact is defined a priori. The first part of the computer model includes the following steps: (i) an investigation of the propagation of the thermal front with a statement of the sintering region, (ii) the modeling of the powder recrystallization into this region, (iii) the modeling of the microstructure shrinkage. The spontaneous microcracking of intergranular boundaries initiates during the cooling and poling process. Above the Curie temperature, ferroelectrics have a cubic structure and microfractures are caused only by considerable thermal gradients [52]. Here, we omit the effects of elastic anisotropy [122]. Below the Curie temperature, residual stresses appear due to deformation phase mismatches and thermal expansion between adjacent grains [102]. The deformation phase mismatches are the most important causes of FC microcracking as the material cools in this temperature range. Computer models of the microcracking of the intergranular boundaries while cooling are described in References [51, 54, 84]. Below, at the modeling of phase transformations near the crack (Section 4), we take into account only spontaneous microcracking during cooling. In modeling ferroelectric domain switching near the crack (Section 5, below), however, we take into account only spontaneous microcracking during poling.

4. MACROCRACK SHIELDING DUE TO PHASE TRANSFORMATIONS NEAR THE CRACK Macrocrack propagation in ceramics may also be investigated by using graph theory [51]. Here, we consider various fracture mechanisms (intergranular, transgranular and mixed ones) and also macrocrack interactions with microcracks, pores and grain phases. References [54, 84] document the influence of the microcrack process zone on various microstructure and strength parameters, showing a real crack trajectory that corresponds to the minimal trajectory length (see Figure 16). The effective surface fracture energy 0, connected with the fracture toughness Kc through the formula Kc = (2E0)1/2, is found as

0 = Lb/h,

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where E is Young‘s module, L is the summary length of all sites of the macrocrack, b is the grain boundary energy, and h is the sample width. Obviously, for rectilinear intergranular crack 0 = b at the zero initial porosity C p0 = 0. Another crack shielding mechanism we examine is connected with existence of phase transformations caused by residual stresses near a growing macrocrack. The crack growth in the FC is defined by its domain structure and by the existence of many phases. Our investigation of crack propagation through domain boundaries show a need to model this phenomenon and estimate corresponding strength properties for prognosis of the behavior of FC strength and fracture toughness [50].

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Figure 16. Example of macrocrack propagation into FC model sample (a) representation of FC structure fragment in computer, (b) macrocrack propagation in the microstructure (grey). Spontaneous cooling microcracks are shown at the grain boundaries, porosity is also shown (grey).

The influence of domain re-orientation in the vicinity of the crack tip on the fracture toughness also has been investigated using a structure model [102]. Analogous phase transition processes are typical for ceramics reinforced by the partially-stabilized zirconium (PSZ) particles [107]. These materials demonstrate the phase transformation under stress from tetragonal to monoclinic crystal structure. In this case, fracture toughness alterations are caused by energy absorption in the phase transformation (PT) zone surrounding the growing crack tip. It is known that PZT ceramics have two phases in morphotropic transition regions (MTR), namely a tetragonal (T) and a rhombohedronic (R) ferroelectric phase. The phase correlations of the volume fractions are found by the technological heterogeneities and by fluctuations of concentrations of the components. They are caused by electrical fields and mechanical stresses [47]. Stresses in the vicinity of the crack tip in FC ceramic of the MTR boundary composition can initiate phase transformations in the ceramic grains. Here we shall estimate the influence of secondary phase inclusions on the effective surface energy p. We use the energy balance method [12] for qualitative evaluation of crack instability conditions due to presence of the PT zone. Then, for the stressed sample, the energy balance method is written as E + W = J,

(10)

where E, W, J are, respectively, the produced elastic energy, the absorbed energy at the phase transformation, and the difference of the surface energy due to the crack propagation in

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the length 2l. We consider infinite elastic plane single-phase solids with a finite crack of length 2l. This crack grows under monotonously increasing remote stresses . We assume that the phase transformation energy W(l) is equal to sum of the intrinsic strain energy W0 for secondary phase inclusions and the interaction energy of inclusions Wint with elastic field of the crack [42]:



W(l) = W0 С (r)dr + 

 С (r)W

int (r)dr .

(11)



Here  is the transformation zone, and the boundary of this region is given as

R 

K12

2

2 1c

cos 2 ( / 2)[1  sin( / 2)]2 ,

(12)

where C(r) is the secondary phase concentration in , K1 is the stress intensity factor of

} is the vector of polar coordinates of the transformation region boundary at the crack tip, and 1c is the critical stress.

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Mode I, r{r,

Then, we transform the dependence W(l), taking into account that it is determined by using the dependences of the zone size of phase transformations and the secondary phase concentration on the crack length 2l, and on the value of the remote stresses . Thus, it is important to state the physically based definitions of the PT zone boundary and the secondary phase concentration into this zone by taking into account stress state at the crack tip. These estimates of the volume phase correlation for the PZT ceramics are found in conditions of mechanical stress homogeneity and assumes an isotropic distribution of orientation angles of the crystallographic directions into grains relative to the loading direction at infinity [48]. The ceramic grain has rhombohedronic or tetragonal symmetry at uniaxial loading. The choice is defined by maximal projection of the lattice distortion vector on tensile loading direction. For these reasons, it is assumed that the phase transformation condition in PZT ceramics is determined by the tensile stress direction along the elementary cell edge and by exceeding the critical stress 1c, i.e., fulfillment of the non-equality

 1 cos2    1c ,

(13)

where 1 is the maximal principal stress near the crack tip, which is found as

1 

K1 2r

cos( / 2)[1  sin( / 2)] ,

(14)

the angle  is the minimal plane angle between direction of the stress 1 and crystallographic direction {100} at the given point. We assume that the FC has rhombohedronic symmetry. Then the value of 1c depends on the rhombohedronic distortion value of the lattice or on the ceramic composition disposition

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at the phase diagram relative to the morphotropic boundary. In the general case, the direction of 1 does not coincide with edge of rhombohedron because ceramic grains are oriented randomly. The rhombohedronic distortion of the PZT pseudocubic lattice is very small (less than 30 minutes) [49]. Therefore, we consider a simple model, where the piezoceramic consists of distribution of the randomly oriented grains into a cubic matrix before loading. As in Reference [48], the existence of the homogeneous distribution of grain orientations independent on adjacent grains is assumed. This assumption neglects the natural ceramic anisotropy which takes place due to processing. We also assume an absence of texture in the FC sample and use the solution of the problem for a random misorientation of cubes [70, 71]. Then, the distribution density of the misoriented angles for crystallographic direction {100} is defined as

8  2  1 tan( / 2) d , 0     ,  sin    dP( )   5 2  32  0,   , 

(15)

where cos  = 2/3. The secondary phase concentration C(r) is found as 

C(r, ) =

1  1c dP( ) ,   arccos , 3  d 0 1

(16)

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where d is the mean grain size. By combining Equations (15) and (16) we obtain

C(r, ) =

1 d3

   1с  1  1  20 

   1с  1с    8  arccos   1 .  5 2   1  1  

(17)

Then, the phase transformation energy, defined by Equation (11), reduces to the form  R

 R

0 0

0 0

W (l )  W0   C (r , )rdrd    Wint (r , )C (r , )rdrd  A 4 l 2 .

(18)

In Equation (18), the parameters W0 and Wint are defined as [63]

W0 

2d 3 (1  0 )  0 M 2 ,  2(1  2 0 )1  (1  0 ) 1    3(1  0 )( 0  1 ) 

Wint  VM 0 .

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(20)

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Here



1

 2(1  2 0 )  V cos( / 2) , M 0  M 1   , M 3, 3(1  0 )  3d 2r 

2 K1

where i (i = 0, 1) are the compression modules of components, 0 is Poisson‘s ratio, 0 is the shear module, V is the elementary cell volume, V is the difference of the V due to phase transition, the symbols 0 and 1 correspond to the matrix and inclusions, respectively, A is the constant depending on V and also on average quasi-isotropic elastic constants of the inclusions and matrix [115]. It also should be noted that due to Equations (15)  (17), the simultaneous existence of two phases in the PT zone is found by the conditions

1c  1(r)  2.251c.

(21)

1 converges to infinity at the r  0, but in actuality the 1 reaches a certain value of m (m >> 1). If m > 2.251c , then the crack Then, Equation (14) shows that the value of

tip is surrounded only by T-phase initiated by residual mechanical stresses. For simplicity, we consider only the dilatation caused by the volume difference due to mismatches of both phases. By returning to Equation (10), we assume for the elastic energy, the ordinary relationship

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δE 

 2l 2 E0

δl ,

(22)

where E0 is Young‘s module of the matrix. Then, the reduced balance energy equation is written as

2 A lδl  4

 2lδl 2 E0

 4 b δl ,

(23)

For Equation (23) we select the solution that makes  > 0 and decreases down to zero at the l  . The first has the form 1/ 2    2 2 b          2 2 Al    8 AE 0  64 A E0 

1/ 2

.

(24)

As it follows from Equation (24) for a sufficiently short crack with length l, subjected to the condition

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l  lc 

128 A b E02

2

,

(25)

any great external stress does not lead to the catastrophic crack growth. Thus, safe cracks exist in the PZT in the framework of the model as described. Analogous result that correspond to the dislocation plastic zone also was obtained [124]. Diagrams of the crack stability in the presence and absence of phase transformations are shown in Figure 17.

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Figure 17. Crack stability diagrams   l: (1) in absence of phase transformations (Griffith‘s model), and (2) in proposed model.

These results are interesting in a physical sense. In the PT zone, the tetragonal inclusions dispose into denser rhombohedral matrix that allows one to create the compressive stresses causing a macrocrack shielding. In the case of short cracks (l < lc), the energetic treatment demonstrates more advantageous initiation of the phase transformations around the crack tip in comparison with its catastrophic growth at any large stress. These calculations define the higher critical load compared with Griffith‘s formula for crack length l > lc. In particular, the critical load in our model for a crack length 2lc has a factor of 2 . Then, in order to calculate the dependence W(l) from Equation (18), we first use Equations (10), (22), (24) and obtain

 2l[1  (1  lc / l )1/ 2 ]   4 b δl , at the l  lc . δW   2 16 E0 A  

(26)

Thus, W = 4bl, at the l = lc. In the case of l >> lc, we decompose the expression (1  lc/l) in a row on powers lc/l and use three terms of the expansion. By applying algebra, we obtain 1/2

δW 

lc  b δl . l

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Our estimate for Equation (26) is obtained on the assumption of minimal applied stress (Griffith‘s stress) leading to catastrophic crack growth. As it follows from Equations (26) and (27), an increase of the crack length leads to a decrease of the energy expended on the initiation of the phase transformations near the crack. It should be noted also that the estimations above are qualitative since they do not take into account the dynamics of the growing crack. The secondary phase influence on the effective surface energy p is estimated taking into account a small deflection in the crack trajectory. By integrating Equation (27) on parameter l in the integrating limits [lc, L] with the next division of the result on the sample width h, we state finally the equation for p:

p = (lc/h)bln(L/ lc),

(28)

In order to obtain the results of computer simulation, we establish the necessary numbers of the computer realizations on basis of the stereological method [13]. The results of computer simulations of the gradient-sintered PZT with different initial porosity C p0 of the original press-powder compact are presented in Table 1.

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Table 1. Computer simulation results Properties

Initial porosity

k D/ D 0 f 0/b

0 0.00 1.00 0.12 1.25 0.17

p/b

C p0 , %

10 0.02 0.94 0.11 1.00 0.17

20 0.04 0.90 0.10 1.00 0.17

30 0.14 0.84 0.10 0.82 0.16

40 0.28 0.83 0.10 0.88 0.15

50 0.43 0.79 0.09 0.90 0.15

Obviously, the microstructure shrinkage factor k = nop/(nop + ncp + ngr) increases with 0 p

C . Here nop, ncp, ngr is the lattice cell number of the open, closed porosity, and grain phase, respectively. At the same time, a decrease of the grain size D with growth of C p0 compensates for the increase of k. These results show approximately equal values of the closed porosity for all considered values of C p0 . The grain size decrease shows a diminution of a fraction of the spontaneous microcracking during the sample cooling (f = lmc/lgb, where lmc, lgb are the summary length of the microcracked intergranular boundaries and all grain boundaries, respectively). The effective fracture energy 0/b demonstrates non-monotonic dependence on C p0 in absence of phase transformations in the vicinity of the crack tip. The minimum is reached at the value of C p0 = 30 %. Analogous discrepant dependence of the fracture energy on the grain size is noted in Reference [102]. The calculations of the parameter lc give the approximate value of 10 m, which agrees with test data. The influence

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of the phase transformations on the fracture energy p/b is approximately the same for all considered cases of the initial porosity. Thus, they contribute only in a small way to the total value of the fracture energy difference.

5. INFLUENCE OF DOMAIN SWITCHING NEAR THE CRACK ON FRACTURE TOUGHNESS After piezoceramic fabrication (i.e., after sintering, shrinkage, cooling and poling processes), the microstructure contains grains, voids, and microcracks at intergranular boundaries and ferroelectric domain structure. We estimate a microcracking by poling due to domain re-orientations that are not equal to 180 ones (otherwise, residual stresses do not occur). In this case, the spontaneous strains  for different ferrohardness degrees are defined in experiment [27] as I = 14103, II = 11103 and III = 8103 (where indexes I, II and III indicate composition by ferrohardness, mean ferrohardness and ferrosoftness, respectively). Then, the critical length of the cracked boundary lcs can be estimated as [84]

lcs 

24 b , E0  2

(29)

where E0 is Young‘s module of the ceramic, and the facet length l that will be cracked for each composition is determined by criterion, taking into account grain misorientation [35] as

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l / lcs  2 /[1  cos(21  22 )] ,

(30)

where i (i = 1, 2) is the angle between the axis of maximum compression in i-grain and the grain boundary plane. In order to estimate these angles, we use the Monte-Carlo procedure [16]. In order to study the influence of the ferroelectric domain switching on changes in crack shielding and fracture resistance, we consider the fracture of ceramic taking into account the poling, re-poling and wake energy process effects by using Kreher-Pompe‘s model [59  61, 106] based on the energy balance method combined with a suitably simplified domain switching model. For a steady-state crack propagating together with a crack tip process zone of domain switching (see Figure 18), the energy balance is presented as [60]

W  W  Wd .

(31)

Here W = Ga is the macroscopic energy supply defined by applied electrical (E) and mechanical () fields, and the change of internal energy outside the crack tip zone, G = G (, E, a) is the release rate of the strain energy, a is the crack size difference. The applied electric field E is construed to be parallel to the crack front, i.e., the poling direction is also parallel to this front and the release rate of the strain energy is dependent only on the applied stress  perpendicular to the crack plane.

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Figure 18. Steady-state crack propagation with crack tip process zone of domain switching. Electrical field is applied perpendicular to plane x1x2 and tension mechanical stress is directed along axis x2.

The criterion of crack growth is G = Gc, where Gc  (1  2 ) K Ic2 / E0 is the critical release rate of the strain energy, KIc is the fracture toughness, E0 is the Young‘s module measured under constant electric field. Then, W = 2ba is the energy which is available locally at the crack tip and which can be consumed at the creation of new fracture surfaces; Wd = 2hda(wd + ww) is the net energy which is dissipated or stored in the nonlinear process zone around the crack tip, where wd is the energy dissipated per unit volume when the material element is transformed from the state outside the process zone into a state in the wake far behind the crack tip; ww is the energy stored per volume in the wake region due to a possible residual stress state existing behind the crack tip. Thus, Equation (31) of energy balance reduces to the following relationship:

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Gc = 2b + 2hd (wd + ww),

(32)

which is defined as the sum of the extrinsic toughness and contribution (positive or negative) caused by the process zone. Analogous relationships are also obtained for various crack problems, for example in References [10, 60, 61, 103, 105, 106]. The introduction of a simplified description of hysteretic FC behavior based on internal variables that measure the relative frequencies of a set of six representative domain orientations allows one to analyze the relevant processes of each material volume element near the crack tip, namely (i) poling due to electric field, (ii) domain switching under mechanical and electric loads, and (iii) partial re-switching into internal stress state in the wake region [60]. Then, Equation (32) may be presented as

 E  , E , m , me  , Gc  2 b  2hd wd0 f   Ec 

(33)

here E c  ( Ec  Ec ) / 2 , E  ( Ec  Ec ) / 2E c (0  E  1), wd0  1.5 s 0 c , and the relative energy densities are

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I. A. Parinov

m  where

3 s 0 c 3E 0  s 0   , me , 2(1  2 ) c 2P s 0 E c

(34)

s0 s0  c  ( c   c ) / 2 ,  , and P denotes the maximal values of remnant strain 



and polarization, respectively. The parameters

 c ( c ) and Ec (Ec ) are shown in 







Figure 19, which also shows the simplified nonlinear constitutive material laws for homogeneous uniaxial loading of ferroelectric and ferroelastic materials. The curves begin from a non-poled isotropic initial state in a linear regime. The ferroelastic (ferroelectric) transition occurs by achieving a certain coercive stress or electric field. This process continues until saturation is attained at  c or Ec . Unloading from the intermediate state

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results in a certain remnant strain and polarization values below the maximal values depicted in Figure 19. As shown in Reference [60], the function f could be explicitly given and for any material the condition 2/3  f  2 fulfilled.

Figure 19. Schematic (a) non-linear uniaxial mechanical (stress  vs strain ) and (b) electrical (dielectric displacement D vs electric field E) behavior of ceramic.

In order to obtain numerical results, we use some typical results for PZT and BaTiO3 ferroelectric ceramics [56, 86], namely E0 = 70 GPa,  = 0.35,

s0  c = 40 MPa,  = 110-3,

s0 E c = 1 kV/mm, P = 0.05 C/m2, E = 0.5. Estimation of fracture toughness requires

knowledge of the size of the process zone hd. It is known [90] that FC poling causes a concentration decrease in domain boundaries in the poled sample as compared with the nonpoled samples. The structural transition of the twinning type under stress near the crack tip can provoke a reverse process of a rise in the domain boundaries density, but a twinning process zone, formed around a propagating crack, will cause the crack to lengthen. Obviously, the process zone size depends on the composition ferrohardness and the ceramic‘s piezoelectric properties. Moreover, the process zone size also is defined by the crack propagation direction. Note also that the maximum influence is rendered on the crack advancing along residual poling direction. By contrast, along normal direction this structure change is restrained by the stress state near crack tip, i.e., the crack propagation is realized

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without an additional energy loss, and the twinning process zone around the crack does not form. The process zone width hd for different ferrohardness degrees is found using a calculation of the parameter, 0    1, reflecting a degree of stability in piezoelectric composition during de-poling (i.e., the ferrohardness degree) [57]. These results are based on the energy balance of the modeled process and on application of the finite element method to calculate the zone of feasible microstructure transformations at the crack tip. For the PZT ceramic compositions I, II and III, respectively, we obtain the next values [57]: 4hd/D = 0.2 (for  = 0.3), 0.5 (for  = 0.5), and 1.2 (for  = 0.7), where D is the mean grain size independent of the ferrohardness degree. A necessary number of computer realizations were again established on the basis of the stereological method [13]. Then, the square cell size  is selected equal to the critical size lcs for composition with mean ferrohardness. For quantitative comparison, we need the grain boundary energy b of FC without any dissipative contribution connected with domain reorientation. The value of b = 3 J/m2 could be used with this aim [86]. The numerical results are present in Table 2 and Figure 20. Table 2. Computer simulation results Properties

Initial porosity

D/l

0 2.04

10 1.92

20 1.84

30 1.71

40 1.69

50 1.61

1.00

0.94

0.90

0.84

0.83

0.79

9.54 23.84 57.23

8.98 22.44 53.86

8.60 21.51 51.62

7.99 19.99 47.97

7.90 19.75 47.41

7.53 18.82 45.17

s c

D/ D 0

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hd, m

ferrohard ceramic ( = 0.3) mean-ferrohard ceramic ( = 0.5) ferrosoft ceramic ( = 0.7)

C p0 , %

Thus, it follows from Table 2 that the size of the process zone hd qualitatively coincides with the corresponding results obtained for barium titanate ceramics possessing near properties, namely in the theoretical model of the ferroelectric domain switching [60] hd = 25  35 m and in experimental observations [79], where a characteristic maximal half-width of the process zone of about 40 m is obtained. Figure 20 also demonstrates qualitative coincidence with the experimental results [56], again nearest FC with mean ferrohardness. Figure 20 also allows an estimate of the degree of toughening due to ferroelectric domain switching, in comparison with the obtained results with the intrinsic fracture toughness K Ic0 (for examples for barium titanate ceramics K Ic0 = 0.48 MPam1/2 [56]). The numerical results for PZT compositions with various ferrohardness qualitatively also coincide with the results of other modeling [86]. Thus, the toughening mechanism caused by ferroelectric domain switching in PZT ceramics contributes the most in fracture resistance as compared with other toughening mechanisms [84, 85]. At the same time, the related martensitic type transformations in hightemperature superconducting ferroelastics YBCO render very small influence on fracture

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I. A. Parinov

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resistance [90, 94]. This is explained by very small magnitudes, defining a spontaneous strain in the YBCO, compared to corresponding values for partially-stabilized ZrO2 [107] and ferroelectric ceramic BaTiO3 [102], where twinning processes play the most prominent role in material toughening. Crack shielding and crack amplification at the fracture of YBa2Cu3O7-x superconducting ceramic is caused by microcracking due to deformation or/and thermal anisotropy, and by the processes connected with microfracture (in particular, crack branching and crack bridging).

Figure 20. Fracture toughness Kic of ferroelectric ceramics dependent on applied electric field E. Black dots represent the experimental results for barium titanate ceramics [56], the open shapes denote results of computer simulations for PZT ceramics at the

C p0

= 0; triangles represent ferrosoft composition,

rectangles - composition with mean ferrohardness, circles - ferrohard composition.

CONCLUSIONS The methods and models presented in this overview cover the entire range of approaches for investigating toughening mechanisms and fracture resistance related to ferroelectric materials. The discussion involves considerations of microcracking and twinning zones near a macrocrack, phase transformations, crack deflection, branching and bridging, and other similar effects. Knowledge of proper toughening and crack amplification mechanisms allows the estimation of total fracture resistance of FC. With this aim, it is defined a ratio K c(i ) / K c0 or Gc(i ) / Gc0 , where Kc(i ) (Gc(i ) ) is the critical stress intensity factor (critical release rate of the

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strain energy) due to i-th toughening (or crack amplification) mechanism; and K c0 (Gc0 ) is the corresponding intrinsic parameter. The total fracture resistance then is stated in the force and energy approach, respectively, as K ctot / K c0 

n

K

(i ) c

i 1

n

/ K c0 , and Gctot / Gc0   Gc(i ) / Gc0 , i 1

where n is the number of the toughening and crack amplification mechanisms. The computer simulation includes the processes of the FC microstructure formation during sintering, the spontaneous cracking by cooling and poling, the energy changes near propagated macrocrack due to the phase transformations, and ferroelectric domain switching near the crack tip. The models we present of the phase transformations and hysteretic domain switching near the macrocrack tip are based on a general energy balance for cracks subjected to the action of a process crack tip zone. We also estimated, for specific models, parameters of fracture toughness and fracture energy in dependence on the initial press-powder porosity of FC, processing technology, applied mechanical and electric fields. The relatively simple models developed to estimate fracture toughness parameters yielded numerical data qualitatively coinciding with existing theoretical and experimental data for similar compositions. Therefore, the our models of the dissipative processes in vicinity of crack describe relevant energy terms sufficient for first order approximations. Current, powerful finite-element approaches and well-developed finite-element software packages (see, for example Reference [1]) will allow more accurate modeling and investigation of crack interaction phenomena occurring in process zones of macrocracks.

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ACKNOWLEDGMENTS This research was supported in part by the Russian Foundation for Basic Research and Federal Aim Program ― Scientific and Scientific-Pedagogical Personnel of Innovative Russia‖ (2009-2013).

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Chapter 7

ACTIVE AND PASSIVE VIBRATION CONTROL OF AIRCRAFT COMPOSITE STRUCTURES USING POWER PIEZOELECTRIC PATCH-LIKE ACTUATORS (1)

S. N. Shevtsov(1) and V. A. Akopyan1(2) Southern Center of Russian Science Academy, Department of Mechanical Engineering,2 Rostov-on-Don, Russia (2) Vorovich Mechanics and Applied Mathematics Research Institute, Southern Federal University,3 Rostov-on-Don, Russia

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ABSTRACT This chapter presents some theoretical and real-world approaches to design and implementation of aircraft structures smart vibration control that is controlled by feedback and shunted by external circuits power PZT patches. First we consider the problem of vibration reduction in helicopter rotor blades, more particularly, the features of rotor blade dynamics and an approach to ensuring dynamic similarity between full scale and scaled rotor blade. On the basis of this analysis, we deduce the principal requirements for smart vibration control of rotor blades. One of the greatest technical difficulties of rotor blade active vibration damping is the necessity to transmit to a blade a number of high-voltage command signals through the rotated hub. The purpose of our investigation was to decrease the number of control channels while retaining good vibration damping efficiency. This study investigates the optimal type (i.e., bimorph or unimorph), location and sizes of plate-like actuators and sensors attached to the composite spar that bears the bend and twist load. We compare the working modes of active and passive (i.e., shunted by electric circuit) PZT actuators.. Numerical simulation shows that the passive damping mode is efficient in the high frequency range only. On other hand, with active control, the stability of control loop may be lost at some vibrations and feedback parameters. We propose an approach according to which all actively controlled PZT patches are driven at 1

email: [email protected]. 1, Gagarin Sq, 344000, Rostov-on-Don, Russia. email: [email protected]. 3 200/1, Stachki Ave, 344090, Rostov-on-Don, Russia. 2

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S. N. Shevtsov and V. A. Akopyan a narrow frequency band, filtering preferentially on a first eigenfrequencies. All installed shunted passive PZT patches will damp high vibration frequencies, while simultaneously increasing the stability of the control loop. Finally, we present some experimental results obtained for a scaled (1/7) rotor.

ABBREVIATIONS ADC/DAC –Analog-to-Digital/Digital-to-Analog converter AFC Active fiber composite FEM – Finite element method FRF – Frequency response function GA Genetic algorithm MFC Macro-fiber composite ODE – Ordinary differential equation PDE – Partial differential equation PID – Combined proportional-integral-digital control (method of forming feedback signal) PSD – Piezoelectric shunt damping PZT – Piezoelectric transducer RLC – Resistive-inductive-capacitive load UAV – Unmanned Aerial Vehicle

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1. INTRODUCTION Recently, a considerable amount of research has gone into developing a rotorcraft flight vehicle with active control of rotor blades [1-4]. An incumbent part of active rotor blade technology is the sensing/actuation system – that is, the piezoelectric transducers disposed on the controlled flexible composite structure. Such devices are intended to reduce a noise and vibration in the helicopter. The possibility of changing the rotor blade‘s local bend or twist with aid of distributed power actuators is necessary for efficient adaptive vibration control. The earliest engineering solution for active rotor blade design is the use of a high-stroke, onblade piezostack actuator for a helicopter rotor with trailing-edge flaps, along with a dualstage mechanical stroke amplifier [3, 5, 54, 55]. Macro-fiber composites (MFC) , also called active fiber composites (AFC) [2, 4, 6-9], retain most of the advantageous features of early piezocomposite actuators, namely, high strain energy density, directional actuation, conformability and durability; yet these composites also incorporate several new features, of which the most important is the use of high-yield fabrication processes that are uniform and repeatable. The principal components of the MFC include a layer of extruded piezoceramic fibers encased in a protective polymer matrix material and the interdigitated electrodes required to produce electrical fields in the plane of the actuator. In-plane electrical fields allow the piezoceramic elements to produce nearly twice the strain actuation and four times the strain energy density, of a through-plane poled piezoceramic device. These features allow the AFC simultaneous precision positioning and vibration suppression capabilities. But due to technological difficulties of manufacturing piezoceramic fibers, a power piezoelectric

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actuator formed as plate-like patches or layers bonded to the surface of host structure most frequently are used. The discrete piezoelectric plates acting as sensors [10, 15] and power actuators [10-24, 56] have been widely used in structural shape and vibration control and investigated in many previous works. However, the proper selection of the number and location of piezoelectric actuators is critical to the efficient control of structural vibration. Therefore, determination of the optimal placement and number of piezoelectric actuators for ensure the control of the structural shape and dynamic state is an important and interesting issue. One of the limitations of a piezoelectric actuator is the amount of force it can exert. Hence, it is important to optimize the location and size of the actuator so that the required control effort is minimal. Similarly, to obtain good signal-to-noise ratio, sensors should be chosen to provide maximum output for the vibration in the modes of interests. Most previous investigations consider the problems of active vibration damping and shape control of plates [11, 13, 17, 19-22] and beams [12, 15, 23, 24]. Some work [12, 17, 20, 23] study the optimization of the piezoelectric actuator configuration at the limit of their complete number. Apparently, finite element method (FEM) analysis [13, 20, 21, 24] used with Genetic Algorithms (GA) [12, 14, 15, 23] is the most efficient method for determining the optimal location of piezoelectric sensors and actuators of intelligent structures. But this analysis applies to cases where there is no limitation on the placement of the piezoelectric transducers. A. Chattopadhyay and C. E. Seeley [25] consider active vibration damping of tubular structures , which is also studied by L. Librescu, and O. Song in their fundamental monograph [26]. However, both cited works neglect the role of flexibility and strain of tube walls caused by power actuators action. Moreover, in cases where multiple actuators are placed on a flexible structure, the interaction between the surface-mounted actuators affects the local stress distribution. The behavior of interfacial stresses is characterized by X. D. Wang [27] as the shear stress singularity factor. The effects of local stress distribution and the adhesive layers [28, 29] also must be considered for rational placement of actuators. Two types of piezoelectric actuators most frequently used to suppress composite structures active vibration are: unimorphs (one layer transducers) and bimorphs (two layer transducers). A few studies [30-34] present the systematic investigation of piezoelectric bimorph bonded on composite plate. But so far, no consensus exists regarding which types of device are more efficient and in which situations. As helicopter rotor blade undergo intensive twist deformations, a specialized actuation system is needed to produce the required shear strain. This problem was considered by A. A. Bent, N. W. Hagood, and J. P. Rodgers [8] for an antisymmetric [45/0/-45], angle-ply, MFC laminate, and by R. Barrett [35], G. Kawiecki and others [36]. According discussions in [36], torque is generated by extension/compression action of a set of piezoelectric elements surface-bonded to the walls of structural elements. The same set of elements may be used to control torsional, bending and longitudinal vibration. Efforts to develop an effective method of torsional vibrations control are very important because significant torsional or combined torsional/bending vibrations may occur in many aircraft structures, e.g., wing spars and helicopters rotor blades. Works [37, 38] propose novel original devices: a torsional actuator designed to produce large angular displacement and large torque output from the piezoelectric d15 shear response, and a piezoelectric coiled bender.. Energy requirements are important design factors in structural intelligent control applications. Average power, peak power required, and total energy consumed directly

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influence control system design [39-42]. While converting electrical energy into mechanical energy to actuate structure, piezoelectric elements experience a temperature increase due to internal heat generation caused by their mechanical damping and dielectric loss. In practice, if the actuation is operated at system resonance or with a relatively high electrical field, the heat generation may be significant. The temperature rise of piezoelectric elements may result in the acceleration of material aging, degradation and even thermal damage. When the thermal resistant materials (e.g., polymeric composite materials) are used for host structures, the temperature in PZT elements increases quickly and cooling may be needed to avoid possible thermal damage. S.-W. Zhou and C. A. Rogers show that thermal stress increases with the thickness of PZT actuators. The mechanical stress generated by a piezoelectric actuator increases with thickness. [43-45] establish that for an actuator-substrate structure, a thicker piezoelectric patch may be preferable for decreasing the stress concentration. The longer the piezoelectric patch, the lower the interfacial normal stress. The interfacial stress increases as the stiffness of the host materials increases[44, 45]. Thus, the selection of optimal dimensions for the actuator has a number of constraints. At active vibration damping, the control purpose is to suppress the displacement of the flexible smart structure; this is a regulating control problem. V.-T. Liu et al. [48] discuss the use of neural network controllers to implement an adaptive vibration control, but majority authors most often use the following types of control: proportional control (P), integral control (I), and derivative control (D), or all together (PID) [46,47, 49,50]. In [49] a digital PID control algorithm is applied to the active vibration control system of an aluminum cantilevered beam. For implementation in a digital computer, the PID control in discrete time is formulated as a function of the actuation signal on preceding samples, of the error signal, of the sample period, of the integration, and the derivative time. Satisfactory performance was determined by such factors as stability, small steady-state error, and minimal settling time. The sample rate was limited by the performance of the equipment, especially the A/D and D/A converters, but the system needs a minimum bandwidth that is about 30 times greater than the suppressed natural frequency of the specimen. V. Sethi and G. Song [47] also note that to experimental implementation of feedback gains requires a low pass filter in the system so as to avoid excitation of higher modes. This undesirable phenomenon was observed experimentally by S. Shevtsov et al. [51]. These actuators also must be controlled by high voltage transmitted from power amplifiers (so-called piezodrivers) through a rotated hub, which is a very complex design task. Due to these circumstances, some authors [56-67] offer a passive piezodamping concept as a low-cost approach, which in most cases does not require an external energy source or complicated electronics. Passive shunt damping (PSD) was studied on various structures, ranging from experimental beam setups to space truss structures, machining processes [66], and aircraft applications [59, 65, 67], with occasional successful results. The applications of PSD are presented in detail in a review article [62]. PSD always is applied by connecting a passive electrical circuit to the electrode terminals of a piezoceramic patch attached to the host structure. D. J. Inman and A. Erturk in [66] studied and confirmed an efficient application of PSD to the suppression of self-excited vibration of tools at relative high frequencies during a machining process. G. Coppotelli et al. [67] developed and experimentally investigated a small unmanned helicopter with flexible rotor blades equipped by PZT patches working in shunted passive mode.

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All of the works cited basically consider a frequency response (FR) of the host structure with some quantity of piezoelectric patches, and their approach is based on the study of PDE with linked ODE describing an electric circuit. This approach provides FR data in a closed form, but are not efficient for the transient analysis of smart structure dynamics that are caused by nonstationary aeroelastic forces.

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1.1 Our Experiment Using as an example a scaled helicopter rotor with active blades, our work demonstrates the way of power piezoelectric actuators use for suppressing forced aeroelastic vibrations. We did not alter their design concepts for complex control systems such as neural network based; however, we restricted our study to the framework of analogue controls and the key problem of devising a morphing PZT patch for use in such structures. We tried to find the optimal type, placement, and size of actuators and sensors attached to a composite D-spar shaped almost like a box-crossed thin-walled composite tube. All simulation results presented below are preliminary, acquired before the design and manufacture of the 1/7-scaled composite active rotor blade. The length of the scaled spar is approximately 1.2 meters and its dimensions are close to rectangular cross-section 2550 mm2. This scaled spar has considerably higher eigenfrequencies than a full-sized rotor blade (length of 8 m, mass of 90 kg). To obtain a satisfactory controllability, limited by best performance allowable to used digital computer, it was necessary to align though first eigenfrequencies of flexural vibration of a full-sized, scaled spar. Therefore, we investigated the eigenfrequencies of the clamped composite tube with mass attached on a free end, both in static mode and at rotor rotation. In the FEM based numerical experiments, we compared the efficiency of the flexural actuation by bimorph and unimorph actuators (working in d31-mode) bonded to a lateral surface of the spar. We investigated only parallel bimorph actuators where both plates are polarized in the same direction and can be connected in parallel. Such a scheme of bimorph connection was selected because it ensures high possible deflections, eliminating the risk of depolarization. For two types of piezoelectric transducers we compared cantilever tip static deflections and greatest shear stresses in bonding layer connecting a body spar with PZT plate. By changing the placement and number of piezoelectric plates, we achieved a pure bending or torsion deformation. The results of finite element analysis demonstrate considerably smaller efficiency of bimorph actuators in contrast to unimorph actuators bending the tube. Next, we studied the features as a source of electric voltage the RLC-shunted piezoelectric rectangular patch subjected to a harmonic tension-compression force, which provided some preliminary data about the frequency range of efficient vibration damping by chosen PZT patch. This information was then used for setting a smart-beam finite element model implemented using Comsol Multiphysics software. This model was simulated at both harmonic and impulse dynamic excitation of beam tip. FEM model results of the beam PSD show very good agreement with estimates predicted by a simples 1D model of the piezoelectric patch. Finally, we compared the efficiency of passive and active (with feedback) vibration damping of the composite beam with pair of small piezoelectric transducers placed opposite, and demonstrated some test results obtained on the scaled helicopter rotor blade.

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2. FULL SIZED ROTOR BLADE AND SCALED BLADE MODEL: SOME CONSIDERATIONS ON THE DYNAMIC SIMILARITY

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For experimental investigation of the scaled active rotor blade, the composite spar of a tail rotor was used. The full sized blade and the scaled rotor blade are depicted in Figures 1 and 2, respectively. The spar of the scaled blade was formed by winding of unidirectional fiber glass tape at a 60-degree angle to the spar axes and curing the epoxy resin in a mould. In all numerical models the following experimentally measured mechanical properties of spar‘s orthotropic composite material were used: the longitudinal Young modulus 30 GPa, transversal in-plane Young modulus 15 GPa, transversal interlaminar Young modulus 12 GPa, in-plane Poisson ratio 0.75, in-plane shear modulus 14 GPa, interlaminar shear modulus 11 GPa. For the finite element model of the clamped composite spar with and without attached tip mass, both static and modal analyses were performed (see Figure 3). Nonrotating, root-hinged, full-scale main helicopter rotor blade has natural vibration modes collocated sequentially as follows: 1st flexural flapwise – 3.0 Hz, 1st flexural chordwise – 9.0 Hz, 2nd flexural flapwise – 9.0 Hz, 2nd flexural chordwise – 27 Hz, 1st torsional – 78 Hz. Figure 3 shows good agreement in the first two natural modes and their eigenfrequencies for both full-sized and scaled spar with attached tip mass.

Figure 1. Full-sized rotor blades.

Figure 2. Tail rotor blade, its composite spar and studied finite element model.

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Figure 3. Natural vibrations modes and eigenfrequencies of the scaled spar with clamped root end: as originally manufactured (top) and with attached tip mass (bottom).

For our modal analysis of the rotated scaled spar, all rigidities were assumed to be constant along the x-axis. The rigidities are as follows: flexural rigidities EIz, EIy, EIzy and torsional rigidity GJ. For the scaled spar, the rigidities were determined numerically using the

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Saint-Venant approach. A small beam pre-twist is neglected. The beam undergoes three displacements: w and v are out-of-plane (flapwise) and in-plane (chordwise) displacements, respectively, and are associated with flexural vibrations in two directions; φ is the rotational displacement associated with torsion. In applying the Euler-Bernoulli bending equations [52], we consider the small linear displacements but not the axial displacements. With these assumptions, the partial ordinary differential equations, governing the free vibrations of beam rotating with angular frequency Ω around z axis, are expressed as follows:









EI y wxx  EI zy v xx xx  Twx x   2 em x  e1  x  m w   e  0,    2 EI z v xx  EI zy wxx   Twx x  mv   mv  0,    0,  GJ x    2 em x  e1 w   2 m k m2 2  k m21   mk m2   emw  





(1)

and boundary conditions coincide with the clamped root end and the attached weight on the right end whose mass is M, and moments of inertia I My , I Mz , I M , giving the following:

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v(e1 )  w(e1 )  v x (e1 )  wx (e1 )  0,  v ( L)   M v( L); v ( L)   I My v ( L). xx x  xxx EI y EI y .  I M Mz w ( L)   ( L); wxx ( L)    x ( L), w w  xxx EI z EI z   (e1 )  0; GJ x ( L)   I M ( L).

(2)

In equations (1) and (2), the following designations were used: m  beam mass per unit length, e  distance between elastic center and center mass of the beam cross-section;

km ,km1 ,km2  radius of gyration; e1  off set beam root; L – beam length, T (x )  centrifugal force. The conversion of problems (1), (2) to the Comsol Multiphysics standard PDE form with mixed boundary conditions gives:

ea u  d a u     cu  u       u  au  f ,   Dirichlet  hu   r ,  T  Neumann  cu   g  h  ,

(3)

where ea ,da ,c, , , ,a,h, g represent matrices formed from equations (1), (2), and μ – represents a vector of the Lagrange multipliers, an eigenvalue problem for these equations at varying Ω was solved. The numerical results are shown in Figure 4, where the dependency of

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the flapwise and chordwise eigenfrequencies vs the angular velocity of full-sized blade rotation are compared with the same dependencies for the scaled blade spar.

Figure 4. The flapwise and chordwise bending eigenfrequencies vs frequency of beam rotation. Fullsized main rotor blade shown in bold lines, scaled blade spar shown in dotted lines.

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These results show that use of the attached tip mass can ensure only one desired eigenfrequency of oscillation in one direction at a given rotation angular velocity. However, it is possible to get other eigenfrequencies (2nd, 3rd, etc.) by changing the tip mass applied to each specific frequency of rotation. In order to speed-up a further finite element analysis, we simplified the shape of spar at the same parameters of flexural and torsional rigidity. With this simplification, the implemented static (at concentrated load) and modal analysis featured a satisfactory coincidence with the beam‘s dynamic properties.

3. BIMORPH AND UNIMORPH FLEXURE ACTUATORS The numerical experiments described below were performed on a simplified cantilever composite tube with a rectangular cross-section (see Figure 5). We investigated different placement schemes for the actuators, two of which are shown in Figure 6: one for bimorph and the second for unimorph actuators. PZT-5H piezoelectric plates were polarized by d31 mode and have the following dimensions 1  20  45 mm3.

Figure 5. Simplified FEM model of composite spar (left-hand tip is fixed).

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Figure 6. Schemes of two pairs of bimorphs and unimorph actuators placed on opposite walls of a tube.

The first series of numerical experiments was implemented on the basis of a full system equation of electroelasticity, where the coupled behavior of the electrical and mechanical subsystems was considered. However, our numerical experiments showed a possibility of also decoupling the subsystems, which was proved by F. Charette et al. [53]. Therefore, the problems of piezoelectric plates and host structure were solved consecutively. This result also would be correct in cases of linear behavior of piezoelectric transducers and bonding layer, and also at frequencies that are sufficiently low relative to the lowest eigenfrequency of the PZT plate. Figure 6 shows the displacement field on the tube wall caused by the action of the piezoelectric bimorph actuator. The four two-layered plates were placed 100 mm from the clamped end. Each layer of bimorph undergoes a longitudinal static force of about 800 N. Figure 7 shows that the tube walls under the bonded bimorph actuators have a major local flexural strain, but the displacement of the cantilevered tip is only 0.2 mm. This deflection is opposite to what would arise in a solid plate. This undesirable phenomenon was not recorded by previous authors [25, 26] studying thin-walled tubular adaptive structures. Finite element modeling of two pair of unimorph actuators at the same active electric voltage shows a deflection of the console tip of more than 7 mm (see Figure 8). As shown, the action of the unimorph actuator is much more efficient than that of the bimorph actuator action. It is important to compare the maximum stress near the edges of both the bimorph and the unimorph actuators. This stress-strain state can be characterized by von Mises stress, as depicted in Figure 9. It is evident that there is an equal amount of stress in both cases . Thus, already the static analysis displays the inexpediency of use the bimorph actuators to create the bending deformation in tube-like structures. Additional examination of these dynamics (which is out of the scope of the given article) has shown that bimorph actuators can create the local oscillation modes stipulated by wall deformation.

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Figure 7. Static displacements on the tube wall caused by the action of 2 pairs of bimorph actuators, (top: close to actuators‘ placement; bottom: along the tube ).

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Figure 8. Static displacements on tube wall caused by action of 2 pairs unimorph actuators. Top: Close to actuators‘ placement ; Bottom: Along tube ).

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Figure 9. Static mode simulation of Von Mises stress on the interface between PZT-plates and composite tube for bimorph (above) and unimorph (below) actuators.

While reasons are specific for studied structures, but they must be considered. Obviously, any resizing of PZT plates cannot qualitatively influence the results. Therefore, an investigation of torsion deformation by piezoelectric actuators was carried out only for unimorph actuators. The current scheme readily allows creation of a bend in the plane of greatest stiffness. For this purpose, it is enough to change the polarity of applied potentials to piezoelectric unimorph transducers. As shown in Figure 10, the deflection of the free end of the cantilever beam reaches 2 mm. It is easy to understand that the asymmetry of the cross-section can be compensated by adjusting the potentials applied to the adjacent piezoelectric plates.

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Figure 10. Bending of tube axes in the plane of greatest stiffness created by applying opposite potentials to piezoelectric unimorph transducers in one row.

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Thus, the use of unimorph actuators to create bending deformations in tubular structures is preferred. Moreover, installation of two identically poled PZT patches with grounded, bonded surfaces on the opposite sides of the tube or beam and excitation of the PZT transducers by equal electric voltage leads to equal displacement along the actuated tube. This useful deduction allows control of two PZT plate deformations with one electrical signal produced by one control channel.

4. TORSION DEFORMATION PRODUCED BY UNIMORPH ACTUATORS To produce the tube‘s torsional deformation, we used a modified scheme (see Figure 11) previously proposed by R. Barrett [35] and G. Kawiecki et al. [36]. Using the conditions described above, we found the torsional deformation along the tube as shown in Figure 12, that is, a pure torsional deformation at intervals of x-axis between 0.12...0.2 m. All related numerical experiments confirm the validity of this rational approach to the placement of actuators for active damping of a spar's forced vibrations. It is therefore desirable to dispose surveyed elementary modules (which it is possible to term as "flexural" and "torsional" actuators) in loops of strains relevant to forced oscillation modes. The overall dimensions, and thus the performance, of such modules is limited by the size of the flat platforms on the upper and lower surfaces of the spar. Thus, one can assert in the static cases at least, that the use of unimorph PZT actuators is much more efficient in changing the shape of tubelike structures, and particularly, in producing or diminishing both bending and twisting deformation.

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Figure 11. Placement of unimorph actuators to produce torsional deformation in a tube.

Figure 12. Change (in degree) of twist angle along x-axis of tube resulting from the action of four Xplaced unimorph actuators.

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5. HARMONICALLY EXCITED PZT PATCH - 1D MODEL As noted above, the use of a considerable number of plate-like actuators embedded on a structure‘s surface may also produce some difficulties with respect to the control architecture., Each PZT actuator (or actuator pair) must be driven by a separate control channel and all high controlling voltage must be transmitted through a rotated hub, which makes for a complex wiring design. The complexity of the control architecture can be very expensive in both cost and weight, which would be prohibitive for use in a flight vehicle. Further, controls can become unstable in the operating frequency range. Together, these circumstances force researchers to carefully study the passive PZT damping mode at which the cyclically deformed PZT plate is shunted by some electrical impedance. Special attention is given to the resistive-inductive shunting case, as it creates a damped dynamic vibration absorber effect if the shunt inductance and resistance are tuned, respectively, to a targeted natural frequency and a desired electronic damping. Therefore, PSD with a passive resistive-inductive circuit recently has been investigated as a viable alternative to other dynamic vibration absorbers and tuned viscoelastic dampers for use in vibration suppression. Research in [66] investigates basic shunt circuits (resistive, capacitive, inductive and resistive-inductive) and demonstrates their application using an analytical model for a cantilever beam excited by a tip force. A methodology for the reduction of the vibration level in a rotorcraft UAV, based on the use of shunted PZT patches tuned on the target frequencies, is presented by G. Coppotelli et al. [67]. It has been shown by B. Glaz et al. [65] that actively controlled flaps with passive dampers can be used to enhance vibration and performance characteristics of blades at high advance ratios. But all of these studies consider coupled electro-mechanical systems that include both PZT transducer and vibrating host structure to be damped. On the other hand, it is also useful to obtain the control parameters for the PZT transducer itself, to help to estimate the properties of passive PZT patch as a source of electrical energy and mechanical stiffener. Problem also is open on whether possible to tune the parameters of an electrical loading shunt to optimize of PZT patch worked as mechanical energy absorber in amplitude and frequency ranges to be damped in aircraft structure. We deal below with these problems. We also consider a thin PZT plate extended along the x-axis, polarized in z-direction (i.e., perpendicular to the main side), and subjected to harmonic, kinematic excitation along the longest side, as shown in Figure 13. The upper and lower surfaces are covered by a conductive layer whose electrical load can have an arbitrary type (Figure 14). Thus, this piezoelectric plate works as strain sensor in e31d31  mode.

Figure 13. Schematic view of the modeled PZT patch working in

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e31 d 31  mode.

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Figure 14. Generalized scheme of PZT patch connected to electric load.

The constitutive equations of piezoelectricity in stress-charge form are as follows:

σ  c E  S  e T  E ,  D  e  S  ε S  E,

(4)

where σ, S - stress and strain tensors respectively - in matrix notation both 6  1 vectors,

D, E - electric displacement (charge density) and electric field respectively - 3  1 vectors, c E - stiffness 6  6 matrix at constant (zero) electric field, e and transposed e T piezoelectric coupling coefficients for stress-charge form - 3  6 and 6  3 matrix respectively, ε S - electric permittivity at constant (or zero) strain - diagonal 3 3 matrix. For

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simplicity, we assume that our plate is a piezoelectric 1D leg, oriented along the x-axis; and we ignore all transversal effects. The equation system (4) is simplified to

σ x  c11E  S x  e31  E z ,  S  Dz  e31  S x  ε 33  Ez .

(5)

Let the lower surface of the piezoelectric plate is grounded and the upper plane, which has a surface area   l  b , is supported under potential V permanent along the plane. The right tip elongation also varies harmonically according to the following law:

l t     exp it  .

(6)

The long-wave approximation leads to the following equations:

 E  σ x  c11  l  exp it   e31  E z ,  D  e    exp it   ε S  E . z 31 33 z l 

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This appoximation is valid for the frequency range less than that of the first natural frequency of the PZT patch. For the plate considered here, the first eigenfrequency (bending in the plane of minimum stiffness) is about 1 kHz. However, when the plate is bonded to the surface of the host structure, all eigenfrequencies of the composite structure are determined by the larger host structure rather than the small piezoelectric plate. The operational frequency range for the host structure vibration damping ranges from 1 Hz to 150...200 Hz. Thus, our admission is well founded. Amplitude values are noted as E z by E0 , and  x by  0 . If PZT plate is loaded onto an electrical circuit with impedance Z and an electrical current generated by PZT patch can be expressed in the following complex forms:     i  e    exp it   i  ε S  E  exp it     E0  exp it   h .(8) iz  D 31 33 0 z   Z l

Then, we obtain the amplitude of the electrical field as follows:

E0  

ie31  l , S h Z   i 33

(9)

and the voltage is shown in the following complex form

V 

ie31 h l ie31b ie31b ie31b    , S h   Y Y Y i  C   1 S 33 L pz L Cpz  i 33  i Z L ZL h

(10)

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where YL , YCpz - electrical conductivity of the load and PZT plate (capacitive), respectively. With the dimensions of the piezoelectric (PZT-5H) plate l = 5 cm, h = 1 mm, b = 1.5 cm and the amplitude of its tip displacement δ = 0.02 mm, we determine the capacity of PZT plate as C pz  0.01F . Figure 15 demonstrates the dependence of the output voltage on the excitation frequency and load resistance. One can see that, in all cases, the real part of voltage approaches the common limit, Vlim   e31b C pz  230V . Complex current flowing through piezoelectric plate, expressed as

I

ie31bYL YL  iYCpz  .

(11)

Full electric power in the complex form at ideal resistive load, expressed as

1 1 1 e31b  * Wel  V  I *  V V *  YL  RL 2 1  RLC pz 2 2 2 2

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Figure 15. Dependence of complex voltage given by PZT patch on excitation frequency at varied load resistance. Real part - solid lines, imaginary part - dotted lines.

This power has an just active character. At    , the electrical power approaches its limit inf el

W

2  e31b  ,  2 RC pz

(13)

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depending on load resistance, internal PZT capacity, the coupling coefficient, and the patch width. At constant excitation frequency  , the electrical power reach its maximum value

Welm 

e31b 2  2C pz

(14)

at magnitude of resistance equal to 1 Rm  C pz  .

(15)

In this case, a time constant of the electric circuit is equal to inverse value of the applied angular frequency

 RC  RCpz   1 .

(16)

These considerations are illustrated in Figure 16, which shows the influence of the excitation frequency and the resistive load on dissipated electrical power.

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Figure 16. Dependency of the PZT-generated electrical power, which dissipates on the resistive load, on the excitation frequency and varied load resistance. Dotted lines show power limit values given by Equation (13) (above), and peak power values given by Equation (14) (below).

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Now we consider the mechanical properties of a vibratory forced PZT patch. After substituting the expression in (9) into the first Equation (7), we obtain an expression for the viscous-elastic coefficient, where the real part is the efficient elastic module and the imaginary part is a dissipative coefficient:

      2 2  e31C pz  C pz    E e31  i      c11  .  l  RL  2 2  2 2  S  1 S  1  33  2   C pz    33  2   C pz     RL   RL    

(17)

As the frequency  grows to infinity, the additive factor in the expression of the 2 efficient elastic module approaches its limit e31

 33S . This value does not depend on the load

resistance (see Figure 17), and the additive value increases the efficient stiffness of the PZT plate just below 3% of the initial value of the module. The imaginary part of Equation (17) describes the damping process, which reaches a maximum intensity at hold of condition (13), and this damping intensity droningly decrease at   1 RLC pz .



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

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Figure 17. Dependence of elastic and dissipative properties of mechanically forced R - shunted PZT plate on the excitation frequency and varied load resistance values. The thin dotted line on the top plot represents the value of initial longitudinal elastic PZT module

c11E .

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It is also of value to determine the mechanical power of the harmonically deformed piezoelectric plate. A full mechanical power in the plate yields:

W

h  2

2 / 

 0



d dt . dt

(18)

After separation of real and imaginary parts, we obtain

ReW  

2 2  2C pz 1 e31b 2 e31 RL2 h    RL  Wdis , (19)      S 2 2 RL 2  l   33 1  RL2 2C pz 2 1  RL2 2C pz





2 e31C pz RL 2 h     E ImW      c11  S 2 2  l    33 1  RL2 2 C pz



  



R h    E 1 e31 b  RL  RL C pz   Wel0  L Wdis  Wel .   c11  2 2 2 Zc 2 l 2 1  RL  C pz 2

2

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306

S. N. Shevtsov and V. A. Akopyan Obviously, the real part Re W   Wdis shows power dissipated in the electrical circuit

(i.e., compare expressions 12 and 19), and the imaginary part is an elastic part Wel and is composed of two components. The first summand Wel0 represents the ideal elastic power in piezoelectricity, without an external electrical circuit; the second one

RL Wdis is caused by Zc

connection with external electrical load. The second component depends on the relationship between internal and external impedances. It is easy to be convinced by a simple calculation that the ratio

RL Wdis Wel0 does not exceed 0.001 for RL  1...1000  kOhm in a Zc

frequency range up to 10 kHz. It follows from here that the full PZT plate‘s elastic power has a very little dependence on the load resistance. Now consider most common case of arbitrary electric load circuit. For an arbitrary of load YL , one can denote YL  Y1  iY2 , where Y1  Re(YL  YCpz ), Y2  Im(YL  YCpz ) . Then voltage and electric current are

V

ie31b ie31b , I  YL . Y1  iY2 Y1  iY2

(21)

Comparing Equations (10) and (11) with Equations (21), we necessarily conclude that the case of resistive – capacitive load can be considered on the basis of all the equations presented above, with the simple substitution Cpz  C  Cpz  CL . Moreover, all the

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results above retain the validity we found, both qualitatively and quantitatively. The real part of the electrical power (equal to mechanical loss)

 h 2 /  d  Y1 (e31b) 2    dt   ReW   Re , dt  2 Y12  Y22  2 0

(22)

and mechanical stress in the PZT plate

x 

 

2 e 2  CY2  e31  CY1  c11E  S 31 2 i  2  S l   33 Y1  Y2   33 Y12  Y2 2 







  .  



(23)

As before, in last the expression the first summand is an efficient elastic modulus and the second one is a damping factor. The following case of load circuits containing parallel connected active-inductive and active loads (see Figure 18) are most interesting. Here again, we overlook PZT leakage resistance and C denote the sum of external and internal capacitances.

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Figure 18. Combined resistive-inductive-capacitive load.

For this circuit, the complex conductivity is

YLR 

R2  iL 1 and YC  iC .  2 2 R2  L  R1

(24)

After some transformations, we obtain for imaginary part of the full conductance

  L , Y2   C  2 2 R2  L   

(25)

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an expression for resonance frequency

0  If

R2

0 and condition for resonance existence

L  CR22  L R2  R2 C , CL2 =

frequency 0  1

0,

we

have

the

well-known

(26) Thomson

formula

for

resonance

CL . For instance, at R2 = 42 kOhm and chosen resonance frequency ν0 =

200 Hz, we obtain from Equation (26) L = 18 H. The corresponding calculation results for dissipated electrical power and efficient elastic modulus in frequency dependence are presented in Figure 19. One can see that decreasing the shunt resistance R1 leads to expansion and to erosion of resonance peaks. As serial resistance R2 decrease (together with corresponding diminishing of inductance according to (26)) the peak of the power shift to the higher frequencies area. It is simple to deduce that as well as in the case of active resistive load at frequency  growing to infinity, the additive factor in the expression of efficient elastic module approaches its limit 2 value e31

 33S , which does not depend on the load's impedance. But contrary to the ideal

resistive load, the variation of the effective module is not monotonic and has a peak at frequency rather greater than the power's peak.

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Calculations (22) - (23) at different values of load inductance and capacity show that PSD in a low frequency range has relatively low efficiency. The results of numerical calculation (see Figure18) are based on the values of inductance 18 H and 74 H. This very large, heavy inductance cannot be used in aerospace apparatus. Thus, a sizeable increase in inductance does not significantly change these conclusions. Therefore, even using of special electric circuit for a synthetic inductor [67] in order to avoid an unacceptable weight of natural inductance is not a good solution for decrease a low frequency vibration. Due to technological limitations, the use of large capacities is preferable as a means of decreasing resonance frequency in tank circuits. Figure 20 shows similar numerical calculations made earlier that are derived at reasonable values of inductances and with relatively large values for load capacity.

Figure 19. Electrical power dissipated on the electric load (above) and efficient elastic modulus (below) vs excitation frequency at resistive/inductive load.

Qualitatively the resonance behavior of the curves presented in Figure 20 is more expressed than behavior of the curves presented in Figure 19. An increase in inductance and capacity causes a decrease in resonance frequency. A decrease in the shunt resistance heightens the resonance peaks. Obviously, the latter is preferable for the purpose of damping wide frequency range vibration . However, a noticeable damping effect is observed only at rather high frequencies. All surveyed loading variants have a similar maximum damping effect reached at an equality of load impedance and internal reactance of the piezoelectric

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transducer. As the current leakage of piezoelectric patch is very small ( Ripz  50106 Ohm),



this condition is Z L  C pz

1 .

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Now we can make some preliminary conclusions about the efficiency of shunted PZT patches for vibration damping in relatively slender aircraft structures. The most powerful vibrations of such structures as helicopter rotor blades, for example, occur at the first natural modes at frequencies 3…100 Hz , and only these frequencies must be effectively suppressed. Changes in the PZT patch‘s stiffness due to its electrical shunting cannot change local and total stiffness of the host structure to any great degree. Therefore, the natural modes and eigenfrequencies of the vibrating structure are not sensitive to shunted PZT patches action. Noticeable damping effects are likely to occur only at rather high frequencies.

Figure 20. Electric power dissipated on the electric load (above) and efficient elastic modulus (below) vs excitation frequency at resistive/inductive/capacitive load.

These conclusions are applicable to any type of piezoelectric material and to rectangular PZT patches with arbitrary dimension. In order to avoid high interfacial stress between a PZT patch and its composite host structure, it is desirable to choose elastic constants of a piezoelectric ceramic that are similar to the same values of the composite material. Thus, an

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apparent increase in a structure‘s stiffness can be reached due to the action of the PZT patch by distributed local moment, as an internal force that acts against external forces (see section 3 above). The damping effect on the whole structure due to an embedded PSD patch can be enhanced in the low frequency range by use of PZT plates with an enlarged surface, within a range defined by the host structure. Also, by augmenting the number of small PSD patches, it is possible to strengthen a structural damping, but only on high frequencies.

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6. TRANSIENT ANALYSIS OF COMPOSITE SPAR WITH EMBEDDED PIEZOELECTRIC PATCHES – 3D FEM MODEL The study of transient processes in smart structures also is of considerable interest, (Works [26, 27, 62-67] examine only FRFs). For investigation of active/passive composite beam vibration suppression, the two FEM models presented in Figure 21 were used. The full FEM model contains two "flexural" (8 PZT patches), as well as two "torsional" modules (also 8 PZT patches). The simplified FEM model consists of two PZT patches placed along a beam axis (one bending actuator). These actuator patches were attached to the upper and lower surfaces of a composite spar. The bonded surfaces of the transducers were grounded, and the poling vectors oriented identically, upward. Due to the identical poling orientation, it is possible to control the bending moment using a pair of PZT patches placed in opposition, by one driving potential. Thus, it is possible to reduce twice as many control channels. In passive mode, each opposite transducer generates an identical potential (that is, if the spar‘s cross section is adequately symmetrical). All numerical experiments described here were performed using the finite element package Comsol Multiphysics in the Structural Mechanics - Piezoelectric mode. This mode includes a full system equation of anisotropic elasticity and electro-elasticity, where the coupled behavior of electrical and mechanical subsystems is considered. In a simulation of passive piezoelectric shunt damping, the electric potentials V1 ,V2 generated by patches were determined from equations, solved for each integration step:

Vi  Ri  J  ds , i  1,2,

(27)



where Ri - shunt resistance, and integrals that express the currents passed through shunts were evaluated from electric current density J normal to patch surface ds . In a simulation of active piezoelectric damping in a simplified beam model, the electric potentials V1 ,V2 supplied to patches were calculated from feedback equation

Vi  k1 x  k2x ; i  1,2 , where deformation

(28)

 x  0.5 1x   x2  and its velocities x  0.51x  x2  were averaged

by integration close to points P1, P2 on the surface of beam close to piezoelectric patches.

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Figure 21. FEM models for transient analysis of active/passive vibration damping. Top – full model of composite tube with two "flexural" and two "torsional" PZT modules; Bottom – simplified model with one pair of PZT patches.

To ensure an optimal dynamic state of the tube FEM model (see Figure 21, top), that is, to minimise the amplitude of both the flexural (in xz and xy-planes) and the torsional (around of x-axis) oscillations in any arbitrary instant, there is a corresponding minimum of the functional:





    xz x, t  x   xy x, t  y   xy x, t   dx , 2

2

2

(29)

L

where the integration is spread spanwise over the beam,

 xy x,t , xz x,t  - the normal strain

on exterior surfaces of the tube at its bending in planes xy and xz , respectively,

 xy x,t  -

strain of torsion around of x - axis also measured on a exterior surface of the tube, and

x x,y x, x - some weight functions. Certainly, the limitations on ultimate electrical

potential applied to piezoelectric plates and mechanical stresses must be taken into consideration. For "measuring" the strain on the surface of the modeled tube, 5 groups of 4 points, placed in vertices of a square with the side length 1 cm (see Figure 21) were created. For each step of the simulation, the program evaluated coordinates of the points and the distance between them. These numerical experiments reveal that for frequencies not

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exceeding 5 first frequencies of flexural oscillations, the flexural strain on the surface of a tube close to piezoplates installation depends linearly on changes in distances between marked points:

 xzi ~ l i  r i  l i1  r i1 ; xyi ~ l i  r i  l i1  r i1  ,

(30)

and the nonlinearity does not exceed 3 %. In expression (30), the Δ indicates increment, a superscript used in distances indicates the number of checkpoints in the group, and a superscript with strain – indicates the number of actuators in the group. Similarly, the torsional strain on the tube surface linearly expressed through changes of distances between vertex points on the squares diagonals is

 xyi ~ d1i  d2i  d1i1  d2i1 .

(31)

Thus, the simulation of strain gauges for measuring the bending and torsional strain is complete. If weight functions in expression (29) become delta-functions localized at the central points of an actuator‘s groups with expressions (30), then integral (29) can be rewritten as the sum over all actuator groups 2 2 2 ˆ     ix   iy   xyi  .

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i



(32)



In active mode, expression (32) is the function of the actual values of the strain, which depend on operating potentials. In the passive damping mode, expression (32) is a function only of the strain, if the parameters of shunting circuits are constant. However, in both active and passive cases, expression (32) is a fast varying function of time. Note that we have intentionally omitted consideration of a problem of minimization of a function (32) in a realtime to concentrate on details of the active and passive damping process. Therefore, we consider a simplified control architecture. According to our approach, the control involves application to the PZT plates of electrical potential, whose value should be proportional to compensated strains and whose velocity uses simple PD - controller. Thus, the electrical potential applied to piezoelectric plates represent linear combinations of strains defined by (30) and (31), and their velocities



   p 



i i i i i i i i V xbend , y t    g x  x  g xt  x  g y  y  g yt  y ,



V tw t    q xi  xi  q xti  xi

i

i



 pti  i ,

,

(33)

where the  sign selected depends on the PZT patch position (to the right or the left of xz plane), and the value of the factors g xi , g xti , g iy , g iyt , qxi , qxti , pi , pti is determined by trial and error.

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The clearest to explain our findings is to express the results of the transient analysis obtained on a simple FEM model of a vibrated beam subjected to harmonic or ― hammer-like‖ excitation. By impulse force, we model a rotor vibration source caused by dissymmetry of lift acting on the advancing and retreating helicopter rotor blades. We concentrate attention on the simple model simulations results. Some results for a beam impulse excited by tip force at different shunt resistances are shown in Figure 22. As shown, the maximum damping effect is observed at shunt resistance close to the internal reactance of the piezoelectric plate. In the case of active impedance presented, maximum damping effect is a value between 100 kOhm and 1 MOhm. At these values, the decrease in rate of vibration amplitude, elastic energy and dissipated electric power is the most intense. In addition, careful observation of the change in vibration spectrum during transient damping process shows that the damping rate of each vibration mode strongly depends on the frequency of these natural modes. This dependence is controlled by the relationship between piezoelectric internal reactance and load impedance, as described above. In order to increase the intensity of vibration damping at low excitation frequencies included in the frequency band 5…30 Hz, we investigated active vibration modes with feedback at different values of parameters k1 , k 2 (see Equation 28). The first parameter modifies an equivalent dynamic stiffness of the host structure, whereas second parameter influences structural damping. Both parameters depend on vibration frequency. Our numerical experiments establish that global structural damping of an investigated beam is less sensitive to k1 variation than to k 2 variation. This phenomenon may be explained by the

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smaller dimension of the PZT patches relative to host beam. However, a second parameter k 2 renders an essential influence to the damping process, increasing an apparent dissipation of material. Unfortunately, compared with a case of passive damping, the active damping mode is characterized by low stability and may be inclined to self-excitation. The passive scheme simply does not exhibit this tendency. In summarizing all results of our transient FEM analysis, we offer the following conclusions. Active damping mode is effective for suppressing stationary monochrome beam oscillations. Feedback parameter k2 (proportional to strain velocity) renders essential influence to the damping process, increasing an apparent local dissipation of material. At suppression of second and third eigenfrequencies, an oscillation on the first eigenfrequency is created. At large k2 values, an instability and self-excitation of the control circuit may originate. Due to phase change, a frequency dependent selective vibration damping/ amplification has been observed. Passive shunt damping is inefficient for suppressing low frequency vibration. Damping effectiveness is enhanced at increasing excitation frequencies. shunt damping even at high frequency requires very big inductance and (or) capacity. Absolute stability in the wide frequency range at different excitation amplitudes is always inherent for PSD. These basic corollaries confirm information obtained in inferences about the area of effective PSD application described in section 4, above.

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314

Figure 22. Passive piezoelectric damping of vibrated beam (Excitation tip force shape is embedded in the mean graph).

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Figure 23. The simplified structure of the scaled active rotor.

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7. TRANSIENT ANALYSIS OF COMPOSITE SPAR BLADE WITH EMBEDDED PIEZOELECTRIC PATCHES –SCALED PHYSICAL MODEL All simulation results were examined in experimental testing of a periodically excited small composite spar of a helicopter tail rotor embedded with PZT patches (see Figure 23). These spars (see Figure 24) were equipped a tip mass attached to ensure the dynamic similarity of both the rotated, full-sized helicopter main rotor blade and of a scaled blade at identical rotation frequencies.

Figure 24. Composite spar of the scaled rotor with bonded piezoelectric patches (all electrical connections are intentionally deleted).

In a static case, the first two vibration modes (flapwise and chordwise) of the scaled blade with attached tip mass have eigenfrequencies between 3 Hz and 10 Hz. As the rotation frequency decreased to its operational limit, the first vibration frequency of the most powerful natural modes reach 10...12 Hz. This frequency range is where the most intensive vibration suppression is needed. For the rotation of our scale model, a rotor vibration source caused by a dissymmetry of lift acting on the advancing and retreating blades was created using two

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electromagnets installed close to the blade tips. At each approach of blade tip to these electromagnets the rotor blade undergoes a vertical shock and vibrates preferentially on low natural frequencies. In article [67] for an investigation of passive piezoelectric vibration damping, the operational frequency ranges over 1 kHz. In a harmonic analysis of the UAV's slender rotor blade with attached PZT patches loaded on RL circuits, the authors [67] report very effective vibration reduction data at 58 Hz. In our study, we tried to perform a quantitative estimate of active damping and PSD for available rotor blade vibration suppression at harmonical and impulse tip loads. All electric signals from piezoelectric and strain gauge transducers were digitized using a multichannel MF624 Humusoft ADC/DAC adapter, working under a Real Time Toolbox soft module implemented as a submodule of the Simulink MATLAB soft package. Thus, all signal handling operations were supported by Simulink program equipment (including filters, phase delay, amplifiers, spectrum analyzers, scopes, etc.) functioning in real-time (i.e., accelerated) mode. All control signals also were formed by this scheme. Two examples of the simplest symbolic schemas are shown in Figure 25.

Figure 25. Examples of symbolic control modules used at testing of active/passive vibration suppression system.

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Our experiments confirm that active vibration damping is more efficient for harmonic excitation, whereas passive mode damping may be successfully used to suppress large spectrum oscillations caused by impulse loads. In a working active suppression system, a low frequency vibration beat usually appears, as illustrated in Figure 26, where time charts of a bending strain (in yellow) and normalized feedback voltage (in magenta) are shown. The beat vibrations with frequency near 0.1 Hz (a) and near 0.4 Hz (b) originates after the feedback switch is turned on. These beat vibrations can be suppressed by changing the parameters k 2 and filtering. We used digital bandpass filters created by Filter Design Toolbox MATLAB, with the following parameters: attenuation on the first and second stopbands – 30 dB, and edges passbands 2 Hz and 20 Hz at FRF nonuniformity 5 dB. The highest magnitude of attenuation was not used because of problems with the computational system; however, these filter parameters ensured satisfactory results (see Figure 27).

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(a)

(b) Figure 26. Active suppression of beam bending vibration caused by harmonical excitation on a frequency near 1st bending natural mode (Feedback switched on at time 1.5 sec). Bending strain shown in yellow, normalized feedback voltage shown in magenta.

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(a)

(b) Figure 27. Enhancing active suppression of beam bending vibration (primary variant see Figure 26, b) (a) feedback coefficient k 2 enlarged twice; (b) the passband filter is included. Bending strain shown in yellow, normalized feedback voltage shown in magenta.

At impulse beam tip load forcing a large vibration spectrum, an optimal selection of feedback parameters k1 , k 2 is not succeed for all excited vibration frequencies. Moreover, the high level feedback may activate nonstability conditions and also the self excited mode of control circuit. For feedback control, use of strain gauges is preferable to piezoelectric sensors, as PZT sensors reinforce high frequency oscillations and impair the control stability. For wide-spectrum vibration damping, the passive, shunted mode of power PZT patches working as high frequency dampers is more efficient. Figure 28 presents time charts of passive vibration suppression . Increasing the number of patches enhances the damping performance capability. Thus, with 4 switched piezoelectric plates placed near blade root (see Figure 24), the damping factor is increased twofold. When the 4 further passive patches are placed on the beam on the distance 30 cm from the beam root the average damping factor increases by 20 to 25%.

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Figure 28. Passive suppression of beam bending vibrations caused by impact load on a beam tip Left undamped vibration (electric potential from open-ended transducers), right - damping by using two opposite placed piezoelectric patches shunted on the 500 kOhm resistance.

For the practical purpose of suppressing flexural-torsional vibration, it is expedient to use a combined active/passive mode. For our small test rotor blade (see Figure 24), we used an active connection and filtering for 4 PZT patches placed near blade root; the ― bending‖ module away from the blade root was connected to an RC shunt. When the frequency of the first torsional mode was high enough (more than 300 Hz), both of the two ― torsional‖ modules were also shunted. With this combined vibration damping architecture and rational choice of feedback factors k1 , k 2 , the vibration amplitude of the free blade tip was reduced from 5 cm to 5-7 mm for 1/2 turnovers. The rotation speed of the rotor thus reached 3 Hz. Nonstability of the control loop and beatings were completely eliminated. At a given performance of the digital subsystem (ADC/DAC, computer, soft), the proposed solution appears to be optimal. However, this method also can be successfully used in more complex, full-scale aircraft smart-structures, as passive electrical components are more reliable than sophisticated digital devices in all cases.

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CONCLUSION The functional features of active, distributed power actuators, on the one hand, and shunted PZT patch actuators for beam like vibration suppression in aircraft structures are investigated. Using a scaled helicopter rotor blade, we present a procedure for building smart aircraft structure equipped by both active controlled and passive shunted PZT patches embedded in the structure to suppress aeroelastic oscillations. Preliminary modal analysis was performed to determine the frequency range of the most intensive vibration. Next, we justified the choice of structural implementation and mounting condition of PZT plates to effectively create bending and torsion, and also to counteract the exterior loadings that create these strains. Our FEM simulations demonstrate considerably smaller operational efficiency of bimorph actuators in contrast to unimorph actuator plates to bend the tube. Therefore, the X-shape-oriented unimorph actuators are offered as an effective way to create torsional strain on the tube. The proposed sensing/actuation scheme for damping of flexural and torsional vibrations is efficient and allow a decrease in the number of control system channels and, for that reason, essentially speed up the response of the digital control. Frequency response functions (FRFs) were measured to show mechanical stiffness and energy dissipation of a shunted, relatively small PZT patch on a simplified 1D model. These FRFs show the relatively low effectiveness of PSD patches for the purpose of changing the dynamic properties (natural modes and eigenfrequencies) and structural damping of the host structure in the low frequency range. FE transient analysis of beam structure with bonded PZT patches controlled by feedback that formed as linear function of local strain and strain velocity of the structure establishes that such an active working mode can be stable in the narrow frequency band. Moreover, due to nonlinearity of a coupled electro-mechanical distributed control system, the suppressed high-frequency vibration can be transformed into vibration on a main form. A combined approach was proposed to effect satisfactory vibration suppression parameters. In this approach, all active controlled PZT patches must be actuated within a narrow frequency band that preferentially filters on a first eigenfrequencies. Thus, all shunted passive PZT transducers installed will damp high vibration frequencies, simultaneously raising the stability of the control loop. All simulation results and subsidiary deductions were tested and confirmed by experimental investigations performed on a 1/7 scaled helicopter rotor.

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[53] Charette, F.; Berry, A.; Guigou, C. J. Intelligent Material Systems and Structures. 1997, No 8, pp 513-524. [54] Grohmann, B.; Maucher, C. K.; Janker, P. In Proc. 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf., 7 – 10 April 2008, Shaumburg, Il, 2008, p 10. [55] Janker, P.; Claeyssen, F.; Grohmann, B.; et al. In Proc. 11th Int. Conf. on New Actuators (ACTUATOR 2008). Bremen, Germany, 9 – 11 June 2008, pp 346-354. [56] Wang, J. In Proc. of the FEMLAB Conference 2005, Stockholm, 2005, pp 153-156. [57] Hagood, N. W.; von Flotow, A. J. of Sound and Vibration. 1997, Vol 146, pp 243-268. [58] Wu, S. Y. Proc. of SPIE. 1998, Vol 2720, pp 259-269. [59] Lesieutre, G. A. The Shock and Vibration Digest. 1998, Vol 30, pp 187-195. [60] Wu, S. Y. Proc. of SPIE. 1998, Vol 3327, pp 159-168. [61] Fleming, A. J.; Berhens, S.; Moheimani, S. O. R. IEEE/ASME Trans. Mechatronics. 2002 Vol 7, pp 87-94. [62] Moheimani, S. O. R. IEEE Trans. Control Systems Technology. 2003, Vol 11, pp 482494. [63] Agneni, F.; Mastroddi, G.; Polli, M. Computers and Structures. 2003, No 81, pp 91– 105. [64] Zhang, J.; Smith, E. C.; Wang, K. W. J. American Helicopter Society. 2004, Vol 49, pp 54–65. [65] Glaz, B.; Friedmann P. P.; Liu, L. In Proc 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf., 7 – 10 April 2008, Shaumburg, Il, p 18 [66] Erturk, A.; Inman, D. J. In Proc. of the Int. Conf. on Noise and Vibration Engineering (ISMA 2008), Leuven, Belgium, 2008, pp 193-206. [67] Coppotelli, G.; Agneni, A.; Balis Crema, L. In Proc. of Int. Conf. on Noise and Vibration Engineering (ISMA 2008), Leuven, Belgium, 2008, pp 157-168.

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INDEX

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A absorption, 194, 244, 268 acceleration, 41, 288 acceptor, 32 accounting, 118, 124, 138, 154, 223, 266 accuracy, 24, 91, 94, 118, 154, 194, 214 acid, 221 acoustic, x, 2, 17, 41, 49, 64, 77, 107, 114, 137, 138, 146, 165, 172, 177, 179, 193, 194, 196, 197, 198, 199, 203, 220, 221, 222, 223, 224, 226 acoustic emission, 2 acoustic waves, 49, 77, 193 activation, 214, 215 active fiber composite, 286 actuation, 286, 287, 288, 289, 320 actuators, x, xii, 72, 75, 109, 285, 286, 287, 288, 289, 293, 294, 295, 296, 297, 298, 299, 300, 312, 320 ADC, 286, 316, 319 additives, 22, 49, 264 adiabatic, 118 adjustment, 73, 138 administration, 109 adsorption, 130 aerospace, 41, 308 AFC, 286 Ag, 139 agents, 138 aging, 288 aid, 109, 286 air, 81, 100, 150, 226 aircraft, 285 Al2O3 particles, 139, 140, 145 alcohol, 24 algorithm, xi, 165, 206, 207, 219, 286, 288 alkali, 4, 22, 23, 24, 50, 51, 53, 55, 56, 57

alkaline, 21 alloys, 220 alternative, ix, 1, 3, 49, 57, 207, 300 aluminum, 288 amorphous, 115, 137 amplitude, 41, 64, 118, 145, 186, 193, 202, 225, 226, 300, 302, 311, 313, 319 anaerobic, 130 angular velocity, 212, 293 anisotropy, 3, 41, 61, 64, 72, 75, 78, 108, 114, 122, 136, 221, 246, 248, 249, 251, 258, 261, 267, 270, 278 annealing, 53 anomalous, 12 antimony, 53 APC, 133 arithmetic, 203 aspect ratio, 81 assumptions, 292 asymmetry, 297 asymptotic, 214, 254 atmosphere, 9 atoms, 15, 18 availability, 229 averaging, 76, 100, 127, 140, 199, 200

B backscattering, 152 bandwidth, 114, 145, 288 barium, 277, 278 basic research, 109 beams, 287 behavior, 5, 12, 15, 16, 33, 36, 38, 43, 45, 46, 61, 64, 85, 87, 133, 142, 149, 151, 186, 220, 226, 253,

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326

Index

258, 260, 262, 264, 266, 268, 275, 276, 287, 294, 308, 310 bending, 21, 245, 261, 287, 289, 292, 293, 294, 298, 302, 310, 311, 312, 317, 318, 319, 320 bias, 58, 61 bifurcation, 258 bismuth, 116 blocks, xi, 8, 61, 177, 179, 207 bonding, 289, 294 bonds, 20, 21, 24, 27 boundary conditions, 76, 77, 96, 120, 138, 178, 181, 182, 183, 184, 185, 187, 189, 190, 193, 194, 195, 198, 207, 208, 209, 213, 224, 230, 234, 235, 236, 237, 238, 239, 240, 292 bounds, 90, 126, 255 branching, xi, 243, 244, 252, 257, 258, 259, 278 breakdown, 266 broad spectrum, 48 broadband, 114 burning, 52, 123, 134, 221

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C CAD, 218 cadmium, 4 calcium, ix, 1, 51 candidates, 75 capacitance, 41 carbon, 123, 130 carbonates, 24, 50, 52, 53, 57, 123 casting, 221 catalyst, 116 cathode, 65 cation, 20, 21, 22, 23, 27, 34 cell, x, 5, 8, 12, 13, 14, 15, 16, 24, 30, 34, 37, 73, 75, 76, 97, 102, 108, 113, 122, 123, 128, 129, 130, 131, 172, 264, 267, 269, 271, 273, 277 channels, xii, 8, 285, 310, 320 charge density, 301 chemical etching, 123 chemical interaction, 138 chlorine, 4 classes, 126, 178, 220 classical, 183, 190, 198 classification, 75 closure, 183, 184, 188, 195, 264 clustering, 4 clusters, 129, 225 Co, 173, 197, 218, 242 codes, 241 coherence, 245, 266 combined effect, 22 combustion, 3

compatibility, 146, 172 complexity, 300 compliance, 119, 143, 151, 222, 251 composition, x, 2, 4, 5, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 22, 23, 24, 30, 33, 34, 37, 38, 39, 49, 52, 57, 58, 64, 72, 73, 87, 115, 122, 134, 137, 138, 163, 268, 269, 274, 276, 277, 278 compounds, ix, 1, 3, 4, 18, 49, 50, 52, 56, 73 computation, 178, 253, 264 computer mouse, 229 computer simulations, 244, 273, 278 computing, 216, 220 concentration, 11, 18, 24, 25, 27, 32, 33, 36, 38, 42, 46, 50, 52, 74, 148, 149, 150, 151, 152, 220, 245, 247, 248, 259, 264, 265, 269, 270, 276, 288 conception, 266 condensation, 18, 20, 21, 205 condensed media, 4 conductance, 307 conduction, 33, 121 conductive, 53, 300 conductivity, 27, 33, 52, 58, 61, 129, 267, 302, 307 conductor, 129 configuration, 215, 220, 256, 287 conformity, 24, 121, 129, 130 conjugation, 179 connectivity, x, 71, 72, 75, 81, 100, 101, 107, 108, 113, 121, 122, 123, 124, 126, 127, 128, 129, 130, 134, 145, 155, 156, 172, 223 connectivity patterns, x, 71, 75 consensus, 287 conservation, 209 consolidation, 221 constraints, 208, 288 construction, 209, 226, 266 consumption, 244 continuity, 76, 131, 150, 188 control, xii, 2, 72, 220, 221, 228, 259, 261, 285, 286, 287, 288, 289, 298, 300, 310, 312, 313, 316, 318, 319, 320 convection, 216 conversion, 22, 292 cooling, 17, 52, 53, 244, 248, 252, 267, 268, 273, 274, 279, 288 cooling process, 244 coral, 122 correlation, 18, 20, 27, 269 correlations, 222, 268 costs, 114 couples, 259, 261, 262 coupling, 2, 3, 17, 21, 24, 38, 41, 48, 64, 72, 73, 97, 99, 102, 106, 118, 121, 131, 132, 133, 136, 143,

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Index 144, 145, 150, 186, 193, 196, 197, 260, 261, 262, 301, 303 covalent, 24, 27, 36 crack, xi, 243, 244, 245, 246, 247, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 271, 272, 273, 274, 275, 276, 277, 278, 279 cracking, xi, 135, 136, 138, 147, 243, 244, 248, 249, 250, 252, 253, 279 critical stress intensity factor, 279 critical value, 129, 252, 253, 254, 255, 260, 264 crosstalk, 146, 172 crystal structure, 27, 36, 268 crystalline, 17, 134, 135 crystallites, 115, 137 crystallization, 17, 19, 42, 267 crystals, ix, x, 9, 17, 19, 20, 71, 113, 115, 138, 162, 163, 171, 220, 266 curing, 290

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D damping, x, xii, 137, 145, 146, 148, 172, 177, 180, 183, 186, 193, 198, 203, 212, 223, 229, 285, 286, 287, 288, 289, 298, 300, 302, 304, 306, 308, 309, 310, 311, 312, 313, 314, 316, 317, 318, 319, 320 decomposition, x, 123, 134, 177, 201, 210 decomposition temperature, 134 decoupling, 294 deduction, 298 defects, 23, 52, 248, 253, 255, 260 deficiency, 8, 9, 10, 15 definition, 192, 231, 245, 247, 266 deformation, 21, 42, 48, 64, 216, 245, 246, 248, 253, 254, 258, 260, 261, 262, 264, 267, 278, 289, 294, 297, 298, 299, 310 degradation, 22, 53, 244, 288 degrees of freedom, 196, 198, 204, 205, 208, 209, 210, 224 delocalization, 20, 21 density, 17, 18, 21, 22, 27, 36, 37, 41, 45, 48, 51, 52, 61, 64, 114, 116, 120, 139, 142, 149, 151, 180, 182, 190, 193, 221, 222, 223, 229, 230, 233, 249, 251, 252, 254, 258, 270, 276, 286, 310 depolarization, 289 derivatives, 180 destruction, 48, 135, 138 detection, 220, 221, 231 deviation, 9, 16, 247, 249 dielectric constant, 81, 149, 150 dielectric permeability, 52, 226, 241

327

dielectric permittivity, 2, 3, 12, 22, 24, 45, 48, 58, 61, 62, 63, 64, 96, 100, 102, 116, 133, 150, 192, 212, 221, 222, 223, 229, 233, 234, 236 differential equations, 181, 224 diffraction, 5, 24 diffusion, 33 discretization, 230, 231 dislocation, 232, 251, 256, 272 dislocations, 251 disorder, 22, 36, 68 dispersion, x, 12, 52, 53, 113, 124, 143, 163, 170, 172, 246 displacement, xi, 18, 20, 33, 76, 118, 120, 121, 178, 179, 180, 181, 183, 193, 208, 209, 212, 213, 214, 223, 224, 227, 228, 230, 231, 232, 233, 238, 239, 240, 276, 287, 288, 292, 294, 298, 301, 302 disposition, 212, 246, 255, 261, 269 distortions, 18, 21, 22 distribution, 10, 75, 87, 107, 134, 137, 138, 193, 210, 214, 220, 221, 228, 229, 230, 231, 253, 258, 266, 267, 269, 270, 287 diversity, 23 division, 254, 273 domain structure, 45, 61, 72, 73, 95, 108, 266, 268, 274 domain walls, 27, 33, 61 ductility, 261, 262, 263 durability, 136, 220, 286 duration, 9, 52, 259

E elastic constants, 152, 234, 271, 309 elasticity, 192, 222, 224, 233, 236, 240, 255, 256, 310 electric charge, 182, 204, 211, 213 electric circuit, xii, 182, 210, 211, 285, 289, 303, 308 electric current, 306, 310 electric field, 48, 58, 59, 61, 62, 63, 64, 73, 74, 77, 100, 102, 109, 118, 179, 180, 223, 266, 274, 275, 276, 278, 279, 301 electric potential, 179, 182, 189, 190, 201, 204, 208, 209, 211, 213, 215, 223, 224, 225, 226, 230, 233, 310, 319 electric power, 302, 313 electrical conductivity, 58, 129, 302 electrical fields, 268, 286 electrical power, 303, 304, 306, 307 electrical properties, 16, 34, 57 electricity, 129 electrodes, 17, 134, 182, 188, 189, 210, 211, 212, 213, 214, 215, 226, 286 electromagnets, 316

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328

Index

electron, 24, 32, 134, 139, 148 electronegativity, 27, 29 electrons, 23, 32 elongation, 261, 301 emission, 2, 256 emitters, 216 energy, xi, 3, 15, 21, 72, 74, 178, 180, 190, 192, 210, 212, 243, 244, 245, 247, 248, 250, 252, 254, 258, 261, 264, 265, 266, 267, 268, 269, 270, 271, 273, 274, 275, 277, 279, 286, 288, 300, 313, 320 energy consumption, 244 energy density, 258, 286 energy supply, 274 engines, 3 environment, 56 environmental protection, 3 epoxy, 101, 107, 290 equality, 91, 94, 209, 269, 308 equating, 9 equilibrium, 32, 223 erosion, 307 estimating, 233, 234, 235, 236, 237, 238, 239, 240, 244, 248, 259, 264 ethanol, 52 excitation, 212, 288, 289, 298, 300, 302, 303, 304, 305, 308, 309, 313, 317 expansions, 213 experimental condition, 23 external influences, ix, 1, 45, 186

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F fabrication, x, 17, 18, 20, 56, 107, 113, 114, 115, 121, 122, 124, 130, 138, 146, 171, 172, 274, 286 family, 4, 18, 22, 75, 137, 146, 244, 266 feedback, xii, 285, 286, 288, 289, 310, 313, 317, 318, 319, 320 feeding, 55 FEM, x, 113, 115, 130, 131, 133, 134, 137, 154, 155, 156, 159, 160, 161, 162, 172, 177, 178, 179, 198, 209, 210, 213, 214, 215, 219, 220, 224, 286, 287, 289, 293, 310, 311, 313, 320 ferroelectrics, xi, 58, 61, 64, 73, 85, 119, 243, 244, 263, 266, 267 FFT, 146 fiber, 286, 290 fibers, 259, 286 field-independent, 58 filters, 316, 317, 320 filtration, 116 financial support, 241 finite differences, 162

finite element method, 115, 118, 130, 154, 178, 203, 205, 220, 277, 287 flexibility, 138, 146, 172, 287 flight, 286, 300 flow, 129 fluctuations, 268 fluid, 193, 194, 196, 197 fluorescence, 8 fluoride, 85 fractal analysis, 130 fractal cluster, 258 fractal dimension, 130 fractal geometry, 121, 130, 258 fracture, xi, 243, 244, 245, 247, 248, 250, 251, 252, 253, 255, 258, 260, 261, 263, 264, 265, 266, 267, 268, 273, 274, 275, 276, 277, 278, 279 fracture processes, 266 fractures, 260 free radicals, 4 freedom, x, 177, 196, 197, 198, 204, 205, 208, 209, 210, 224, 225 freezing, 131 friction, 261 Friedmann, 323

G gas, 130, 193 gauge, 316 gels, 115 generation, 23, 72, 85, 116, 288 generators, 2 geometrical parameters, 263 glass, 290 glasses, 22 goals, 225 grain, 25, 27, 132, 134, 135, 246, 247, 248, 250, 251, 252, 253, 258, 259, 260, 261, 267, 268, 269, 270, 273, 274, 277 grain boundaries, 135, 248, 253, 268, 273 grains, 27, 131, 248, 249, 251, 252, 253, 259, 261, 267, 268, 269, 270, 274 grants, 216 granules, 123, 146, 147, 148, 149, 150, 151, 152, 153, 172 graph, 267, 314 graphite, 130, 221 groups, 2, 64, 73, 115, 137, 205, 207, 311, 312 growth, xi, 20, 21, 55, 61, 65, 133, 134, 135, 145, 151, 243, 244, 245, 247, 252, 253, 254, 255, 258, 259, 260, 261, 264, 266, 267, 268, 272, 273, 275 gyroscope, 210, 211, 212, 214, 215, 216

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Index

H

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hardness, 27, 33, 36, 58, 64, 139 harmonics, 124, 145, 170, 171 heat, 134, 216, 267, 288 heat conductivity, 267 heat transfer, 216 heating, 17, 52, 53, 215, 216 height, 245 helicopters, 287 heterogeneity, 226, 245, 246 heterogeneous, 126, 178 high resolution, 134 high temperature, ix, 1, 2, 49, 114 higher quality, 55 high-frequency, 3, 17, 37, 39, 41, 48, 64, 170, 320 homogeneity, 249, 269 homogenous, 183, 195 host, 15, 22, 23, 30, 287, 288, 289, 294, 300, 302, 309, 310, 313, 320 hot pressing, 5, 17, 20, 24 hub, xii, 285, 288, 300 hydroacoustic, 90 hydrocarbon, 222 hydrolysis, 50, 52, 53, 55 hydrophone, 145 hydrostatic pressure, 3, 20 hydroxides, 51 hypothesis, 126 hysteresis, 17, 58, 59, 60, 61 hysteresis loop, 58, 60

I identity, 77, 101, 195, 202, 223 imaging, 75 impedance spectroscopy, 120, 138, 139, 143, 146, 148, 162, 165, 172 implementation, xii, 285, 288, 320 inclusion, 100, 210, 245, 246, 247, 255, 256, 264 indicators, 3, 49 induction, 223 inductor, 308 industrial, 3, 24, 116 industry, ix, 1, 114 inequality, 90, 96, 142, 145, 151 inertia, 181, 189, 191, 292 inertness, 257 inferences, 313 infinite, 64, 129, 138, 248, 255, 258, 269 infringement, 131 inhibitors, 134

329

inhomogeneity, 170 initial state, 276 initiation, 248, 250, 253, 258, 272, 273 injection, 53 inspection, 113, 163 instabilities, 4, 12, 19, 20, 22 instability, 258, 268, 313 insulation, 116 integration, x, 177, 185, 194, 196, 198, 288, 310, 311 interaction, xi, 4, 21, 27, 33, 37, 220, 243, 246, 251, 254, 255, 257, 259, 269, 279, 287 interaction effect, 259 interaction effects, 259 interactions, xi, 109, 243, 244, 245, 251, 255, 267 intercalation, 21 interface, 229, 263, 297 intermolecular, 259 internal combustion, 3 interphase, 38, 258 interstitial, 9, 10, 11, 15, 16, 21, 23, 32 interstitials, 22 interval, 17, 22 intrinsic, 180, 260, 269, 277, 279 Investigations, 259 ionic, 4, 9, 15, 18, 34 ionization, 15 ionization energy, 15 ions, 8, 9, 10, 11, 16, 21, 27, 33, 48, 73 IOP, 78, 79, 80, 84, 88, 89, 91, 92, 94, 95, 97, 98, 99, 101 isomorphism, 4, 11 isothermal, 118 isotropic, 100, 127, 165, 253, 269, 271, 276 isotropy, 248 ISS, 10, 15, 16 iteration, 205

K K+, 15, 21 kinetic energy, 190 kinetics, 37 kinks, 46 knots, 231 KOH, 51

L labor-intensive, 50 Lagrange multipliers, 292 Lagrangian formulation, 224 lamina, 75, 100, 101, 102, 109

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330

Index

laminar, 75, 100, 101, 102, 109 laminated, 100 language, 228 lanthanum, 27 lattice, 5, 8, 9, 10, 11, 12, 14, 17, 18, 21, 23, 24, 30, 33, 34, 129, 248, 267, 269, 273 lattice parameters, 5, 12, 14, 17, 24, 30, 34 law, 4, 129, 143, 186, 192, 250, 301 laws, 276 leakage, 306, 309 lift, 313, 315 limitation, 287 limitations, 38, 114, 118, 122, 155, 287, 308, 311 linear, x, 58, 61, 67, 118, 120, 143, 155, 170, 177, 183, 187, 200, 203, 205, 207, 208, 221, 223, 225, 230, 233, 251, 255, 262, 266, 267, 276, 292, 294, 312, 320 linear dependence, 251 linear function, 320 liquid phase, 22, 32, 51 lithium, ix, 1, 9, 21, 30, 32, 36, 37, 52, 65, 66, 67, 116, 124 loading, 229, 248, 253, 254, 256, 258, 259, 260, 262, 263, 264, 269, 270, 276, 300, 308 localization, 212 location, xii, 285, 287 losses, x, 17, 25, 33, 64, 113, 118, 121, 124, 154, 155, 162, 163, 165, 172 low-temperature, 264

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M M1, 12, 14 magnesium, 114 magnet, 220 magnetic, 24, 248 magnetic field, 248 manipulation, 221 manufacturing, 2, 40, 49, 72, 114, 115, 116, 122, 124, 134, 220, 221, 286 market, 115 martensitic transformations, 263, 266 materials science, 72 measurement, 3, 116, 124, 136, 143 mechanical energy, 288, 300 mechanical properties, 114, 129, 163, 222, 233, 290, 304 mechanical stress, 20, 48, 76, 77, 131, 136, 179, 180, 181, 268, 269, 271, 275, 288, 306, 311 media, 4, 51, 100, 126, 179, 180, 185, 188, 193, 203, 217, 218, 220, 222, 223, 224, 225, 226, 251 medical diagnostics, 41, 113 melt, 17, 19, 267

melting, 9, 51 melting temperature, 51 memory, 266 mercury, 4 mesh node, 231, 232 mesoscopic, 4 metal ions, 73 metals, 51, 55, 56, 73 MFC, 286, 287 Mg2+, 23, 25, 27, 29 microcracking, xi, 243, 244, 248, 249, 250, 251, 252, 253, 254, 255, 258, 259, 267, 273, 274, 278 microdefects, 244, 251 microscope, 17, 24 microscopy, 139, 261 microstructure, x, xi, 113, 122, 127, 129, 133, 134, 138, 139, 171, 220, 221, 222, 243, 244, 246, 247, 248, 250, 255, 259, 260, 265, 267, 268, 273, 274, 277, 279 microstructure features, xi, 243, 247, 259, 265 microstructures, 147, 244 microwave, 17, 22 mixing, 24, 126, 221 mobility, 10, 33, 61 modeling, x, 109, 115, 118, 121, 127, 129, 130, 131, 137, 138, 154, 155, 163, 172, 177, 179, 216, 220, 225, 226, 228, 229, 230, 244, 261, 262, 267, 277, 279, 294 models, x, xi, 113, 115, 121, 127, 128, 129, 130, 131, 137, 155, 165, 166, 167, 171, 172, 178, 180, 223, 226, 243, 248, 255, 256, 258, 262, 266, 267, 278, 279, 290, 310, 311 modules, 216, 229, 230, 241, 246, 248, 249, 250, 251, 253, 271, 298, 310, 311, 316, 319 modulus, 2, 24, 38, 43, 52, 58, 59, 131, 149, 150, 170, 212, 222, 290, 306, 307, 308, 309 moisture, 52 morphological, 245, 261 morphology, 4, 48, 220, 221, 247 motion, 61, 180, 212 motors, 2 multiplication, 192, 207, 210, 228, 229 multiplicity, 37

N Na+, 9, 10, 11, 15, 20, 21, 23 natural, x, 17, 177, 186, 188, 189, 190, 224, 234, 248, 259, 270, 288, 290, 300, 302, 308, 309, 313, 315, 317, 320 Nb, 4, 20, 21, 22, 23, 32, 73 neglect, 223, 287 network, 129, 193, 288, 289

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Index neural network, 288, 289 Ni, 73, 134 niobium, 24, 33, 36 nitrates, 123 nodal forces, 214 nodes, xi, 178, 204, 208, 209, 213, 214, 226, 230, 231, 232 noise, 3, 64, 78, 286, 287 normal, 73, 76, 126, 179, 181, 193, 197, 208, 247, 276, 288, 310, 311 novel materials, 2 NPI, 71 n-type, 25, 32 nucleation, 267 nuclei, 248 nucleus, 250

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O observations, 258, 277 octane, 52 oil, 134, 139 one dimension, 94 operator, 179, 187, 188, 199, 200 opposition, 310 optical, 17, 19, 24, 116, 134, 135, 139, 165 optical micrographs, 116 optical microscopy, 139 optical properties, 17 optimization, 36, 137, 138, 154, 155, 162, 172, 215, 220, 244, 287 ordinary differential equations, 292 organic, 134, 220 orientation, x, 45, 71, 73, 75, 76, 78, 81, 87, 90, 108, 122, 245, 246, 247, 248, 255, 256, 261, 266, 268, 269, 277, 310 orthogonality, 188, 206 orthorhombic, 5, 12, 14, 17, 23, 25, 37, 42 oscillation, 41, 52, 216, 226, 228, 233, 293, 294, 298, 313 oscillations, xi, 41, 178, 186, 202, 219, 228, 230, 233, 234, 311, 312, 313, 317, 318, 320 OST, 24, 66 oxidation, 23, 27 oxide, 22, 68, 244, 264 oxides, 2, 8, 18, 22, 24, 48, 50 oxygen, 5, 8, 9, 12, 14, 18, 23, 24, 27, 33, 48

331

P parameter, x, xi, 8, 9, 10, 11, 12, 14, 15, 24, 25, 33, 34, 42, 46, 48, 84, 90, 113, 178, 194, 210, 213, 233, 273, 277, 279, 313 particle shape, 134 particles, 24, 127, 137, 138, 139, 140, 142, 143, 144, 145, 146, 147, 221, 245, 246, 247, 259, 261, 262, 263, 268 passive, xii, 81, 102, 122, 134, 285, 288, 289, 300, 310, 311, 312, 313, 316, 317, 318, 319, 320 Pb, ix, 1, 2, 3, 4, 48, 49, 64, 65, 71, 72, 73, 135, 136, 137 PCR, 16, 49, 105, 106, 107, 124, 125, 132 percolation, 115, 116, 121, 127, 129, 133 percolation theory, 129 periodic, 122, 126 permeability, 52, 226, 241, 245 permittivity, 12, 16, 38, 48, 61, 119, 126, 127, 180, 190, 212, 222, 236, 301 perovskite, ix, 1, 4, 5, 8, 12, 13, 14, 15, 18, 20, 22, 24, 30, 34, 37, 72, 73, 75, 76, 102, 108, 263, 265 perovskite oxide, 22 perovskites, ix, 1, 9, 72 perturbation, 255 PFC, 130, 134 phase boundaries, 4, 12, 126 phase diagram, 4, 5, 6, 8, 11, 12, 15, 17, 18, 19, 37, 38, 41, 42, 45, 270 phase transformation, xi, 4, 243, 244, 266, 267, 268, 269, 270, 272, 273, 278, 279 phase transitions, 4, 5, 14, 16, 17, 37, 266 physical mechanisms, x, 113, 172 physical properties, x, 4, 18, 20, 22, 23, 25, 42, 72, 73, 117, 177, 190, 221 piezoceramics, 122, 126, 241, 282 piezocomposites, 113, 117, 137, 146, 219, 242 piezoelectric properties, xi, 13, 22, 23, 34, 37, 41, 57, 64, 72, 85, 108, 114, 120, 122, 130, 134, 219, 221, 222, 229, 233, 276 piezoelectricity, 3, 72, 118, 173, 178, 183, 222, 301, 306 pipelines, 3 planar, 38, 52, 75, 144 plants, 3 plasma, 50 plastic, 122, 251, 259, 261, 262, 263, 272 plastic deformation, 261, 262 plasticity, 261, 262 platforms, 298 play, 11, 33, 73, 76, 278 PMMA, 122, 221 point defects, 22

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332

Index

Poisson, 212, 265, 271, 290 Poisson ratio, 290 polar coordinates, 269 polarity, 297 polarization, 12, 17, 18, 19, 20, 22, 38, 43, 48, 55, 57, 58, 61, 64, 65, 131, 150, 211, 212, 228, 230, 233, 276 polarized light, 19 pollution, 56 polycrystalline, 115, 116, 117, 119, 124, 137 polymer, ix, x, 71, 75, 76, 77, 78, 79, 81, 84, 85, 87, 88, 90, 94, 95, 96, 97, 100, 102, 105, 107, 108, 109, 113, 115, 122, 123, 137, 171, 220, 222, 286 polymer composites, ix, 71, 75, 78, 79, 85, 90, 96, 122 polymer matrix, 81, 95, 96, 100, 108, 286 polymer structure, 109 polymers, 101, 102, 109, 118 polymethylmethacrylate, 222 polymorphism, 4 polyurethane, 102, 104 pore, 9, 121, 128, 129, 130, 131, 132, 134, 155, 163, 220, 221, 226, 228, 229, 230, 231, 233, 241, 260, 263, 267 pores, 81, 98, 115, 122, 124, 128, 130, 134, 137, 143, 152, 155, 171, 220, 221, 222, 228, 229, 232, 236, 267 porous materials, 124, 129, 130 porous media, 225 porous solids, 130 potassium, ix, 1, 12, 124 powder, 5, 24, 58, 134, 138, 220, 267, 273, 279 powders, 17, 52, 115, 122, 134, 138, 146, 147 power, xii, 3, 49, 129, 249, 254, 285, 286, 287, 288, 289, 302, 303, 304, 305, 306, 307, 308, 309, 313, 318, 320 prediction, 2, 115, 130 preference, 251 press, 267, 273, 279 pressure, 3, 9, 20, 21, 24, 49, 114, 131, 150, 193, 194, 198, 223 probability, 129, 260 production, 2, 49, 50, 51, 55, 114, 115, 122, 124, 134, 136, 220, 221 production costs, 114 prognosis, 268 program, xi, 155, 165, 170, 177, 198, 229, 311, 316 propagation, 16, 17, 245, 246, 248, 250, 251, 252, 254, 257, 259, 260, 261, 264, 267, 268, 275, 276 protection, 3 PSD, 286, 288, 289, 300, 308, 310, 313, 316, 320 PTT, 68 p-type, 33

pulse, 145, 146, 148, 162, 163, 165, 166, 167, 168, 169, 170, 172 pulses, 41, 64, 145, 170

Q quadrupole, 21 quasilinear, 267

R radiation, 5, 24, 52, 58 radio, 220 radius, 15, 34, 100, 245, 263, 292 radius of gyration, 292 random, 138, 270 raw materials, 50, 54, 55 Rayleigh, 17, 124, 138, 143, 170, 180, 183, 188, 223 reactivity, 15 reagent, 5, 24 reality, 119 reasoning, 21 recrystallization, 267 rectilinear, 256, 268 redistribution, 11, 34, 78, 102, 109 refractory, 134, 138 regular, 2, 9, 11, 18, 21, 34, 48, 73, 75, 87, 107, 138, 152, 225, 233, 248 regulation, 55 regulations, 3 relationship, 11, 118, 119, 120, 253, 271, 275, 306, 313 relationships, 2, 4, 22, 37, 38, 223, 251, 275 relative size, 248 relaxation, 12, 73, 119, 266 reparation, 138 repeatability, 54 resin, 290 resistance, xi, 17, 116, 211, 212, 214, 221, 222, 243, 244, 245, 251, 252, 253, 258, 260, 274, 277, 278, 300, 302, 303, 304, 305, 306, 307, 308, 310, 313, 319 resistive, 300, 302, 303, 304, 306, 307, 308, 309 resistivity, 24, 25, 36, 40 resolution, 24, 114, 118, 134, 137, 145, 154, 172 resonator, 120 returns, 25 rhombohedral, 5, 23, 25, 37, 42, 74, 272 rigidity, 291, 293 risk, 289 rods, 81, 94, 95, 96, 100, 101, 102, 106, 108, 150, 154, 155

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Index room temperature, 3, 5, 24, 37, 52, 55, 56, 74, 75, 76, 85, 86, 105 room-temperature, 5, 17, 105 rotational transformations, 18 rotations, x, 131, 177, 208 routines, 165

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S sample, 24, 52, 57, 58, 76, 78, 81, 87, 94, 100, 102, 107, 120, 121, 127, 131, 138, 139, 140, 150, 247, 248, 249, 258, 267, 268, 270, 273, 276, 288 saturation, 251, 253, 266, 276 scalar, 183, 184, 185, 188, 192, 195, 206 scatter, 10, 15 scattering, x, 24, 113, 124, 137, 138, 140, 142, 143, 144, 146, 147, 148, 152, 163, 170, 172 schema, 210 schemas, 316 SCs, 72, 73, 74, 75, 76, 95, 96, 107, 109 search, 3, 49, 65, 109 searching, 205, 207 SEM, 123, 132, 139, 147, 148, 164, 165 SEM micrographs, 123, 147, 164 semiconductor, 116 sensing, 116, 286, 320 sensitivity, x, 2, 3, 49, 64, 71, 78, 80, 90, 102, 107, 114, 118, 121, 137, 145, 146, 154, 172 sensor technology, 216 sensors, x, xii, 41, 72, 75, 107, 109, 285, 287, 289, 318 separation, 9, 24, 207, 245, 305 series, xi, 12, 37, 52, 78, 120, 123, 126, 178, 230, 231, 294 shape, 24, 38, 58, 61, 81, 100, 101, 122, 128, 137, 138, 154, 190, 191, 195, 209, 224, 230, 247, 261, 264, 266, 287, 293, 298, 314, 320 shear, 3, 8, 15, 23, 33, 36, 119, 127, 146, 148, 154, 155, 165, 206, 210, 212, 214, 215, 216, 247, 264, 266, 271, 287, 289, 290 shock, 316 shunts, 310 sign, 182, 246, 312 signals, xii, 145, 146, 172, 285, 316 signal-to-noise ratio, 64, 78, 287 similarity, xii, 22, 96, 285, 315 simulation, xi, xii, 162, 163, 165, 172, 233, 243, 244, 258, 273, 277, 279, 285, 289, 297, 310, 311, 312, 315, 320 simulations, 171, 244, 273, 278, 313, 320 single crystals, ix, 17, 21, 71, 72 singular, 255 singularities, 248, 249

333

sintering, ix, xi, 1, 2, 24, 33, 38, 40, 41, 42, 115, 123, 134, 138, 142, 148, 220, 221, 222, 243, 244, 248, 267, 274, 279 SiO2, 22 sites, 8, 9, 10, 11, 15, 16, 23, 25, 32, 34, 36, 248, 252, 268 skeleton, 122, 130, 131, 134, 150, 151, 155, 172 smart materials, 72 smoothing, 140 sodium, ix, 1, 4, 12, 124 software, 115, 121, 130, 139, 146, 148, 154, 155, 162, 165, 179, 212, 225, 228, 229, 233, 236, 241, 279, 289 solid phase, 52, 134 solid solutions, ix, 1, 2, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 22, 23, 24, 25, 27, 30, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 49, 57, 58, 61, 64, 71, 72, 73, 74, 107, 124 solid-state, 2, 5, 24, 37, 38, 57, 72 solubility, 38 sound speed, 43, 45, 48 spatial, 145, 163, 165, 170, 172, 261 species, 8 specific heat, 19 specificity, 179 spectroscopy, 8, 119, 120, 138, 139, 143, 146, 148, 162, 165, 172 spectrum, 45, 48, 52, 120, 121, 125, 140, 152, 155, 172, 188, 313, 316, 317, 318 speed, 193, 224, 293, 319, 320 spheres, 122, 123, 127, 179, 220, 247 springs, 262 sputtering, 65 stability, xii, 2, 3, 17, 18, 20, 45, 52, 61, 272, 277, 285, 288, 313, 318, 320 stages, 52, 55, 56, 100, 115, 179 standards, 66, 118, 154, 155 steric, 34 stiffness, 119, 143, 151, 192, 200, 205, 224, 226, 229, 288, 297, 298, 301, 302, 304, 309, 310, 313, 320 stochastic, 138, 143, 170 stoichiometry, 8, 9, 10, 16, 50 storage, 122 strains, 73, 263, 274, 298, 312, 320 strength, ix, xi, 1, 18, 22, 58, 116, 243, 244, 245, 249, 258, 260, 261, 265, 267, 268 stress fields, 223 stress intensity factor, 248, 250, 252, 255, 269 stretching, 20 stroke, 286 strontium, ix, 1, 50 structural characteristics, 12

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334

Index

structural defect, 12, 22, 33 structural defects, 12, 22, 33 structural transitions, 48 subdomains, 181, 182, 193 substances, 56, 115, 137 substitutes, 15, 32 substitution, 16, 21, 32, 34, 115, 137, 208, 306 substrates, 55 superconducting, 263, 266, 277 superconductor, 266 superconductors, 265 superposition, x, 20, 177, 259 suppression, 21, 41, 286, 288, 300, 310, 313, 315, 316, 317, 318, 319, 320 surface area, 119, 301 surface energy, 268, 273 suspensions, 115, 221 switching, xi, 61, 243, 244, 266, 267, 274, 275, 277, 279 symbols, 19, 212, 271 symmetry, 4, 5, 12, 30, 42, 45, 48, 58, 64, 74, 75, 76, 85, 86, 87, 119, 180, 198, 249, 258, 269 synthesis, ix, 1, 2, 17, 24, 37, 38, 42, 49, 50, 51, 52, 55, 58, 115, 123, 134

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T tantalum, 52 temperature dependence, 53, 216 tensile, 269 tensile stress, 269 tension, 246, 252, 260, 262, 275, 289 terminals, 288 ternary oxides, 8 Tesla BS, 24 test data, 273 thermal analysis, 17 thermal decomposition, 123 thermal expansion, 138, 147, 246, 248, 249, 258, 267 thermal properties, 129 thermal stability, 17 thermodynamic, 18, 19, 20, 22, 118 thermodynamic parameters, 18 thermodynamic properties, 18 thermogravimetric, 24 thermo-mechanical, 248, 260 Thomson, 281, 307 three-dimensional, xi, 122, 129, 165, 178, 210 threshold, 127, 133, 249, 252 thresholds, 116, 129 time elements, 136 titanates, 20 titanium, 50

tolerance, 18, 22 topology, 129 torque, 287 total energy, xi, 243, 253, 287 toughness, xi, 243, 244, 245, 247, 250, 251, 252, 253, 255, 258, 260, 261, 264, 265, 266, 267, 268, 275, 276, 277, 278, 279 toxicity, ix, 1, 3, 49 trade, 114 trade-off, 114 trajectory, 246, 247, 267, 273 transducer, 41, 64, 113, 114, 115, 116, 117, 137, 145, 153, 171, 172, 216, 222, 286, 300, 309, 310 transfer, 204, 216 transformation, 21, 72, 244, 258, 259, 264, 265, 266, 268, 269, 270 transformations, xi, 4, 18, 209, 210, 243, 244, 264, 266, 274, 277, 307 transition, 5, 12, 16, 17, 21, 61, 73, 115, 129, 255, 263, 268, 271, 276 transitions, 17 transmission, 17 transparent, 17 transpose, 119 trial, 52, 312 trial and error, 312 tubular, 287, 294, 298 twinning, xi, 243, 244, 266, 276, 278 two-dimensional, 230, 233, 236, 240, 248, 249, 267

U ultrasonic waves, 124, 143, 152, 170 ultrasound, 116, 165 uncertainty, 121 uniform, 12, 24, 25, 34, 41, 64, 286

V vacancies, 8, 12, 22, 23, 24, 27, 32, 33, 36 vacuum, 134, 212 valence, 8, 15, 23, 24, 32, 33, 73 validation, 115, 154 validity, 121, 155, 298, 306 Van der Waals, 259 variables, xi, 119, 120, 165, 177, 275 variation, 4, 10, 11, 12, 33, 34, 45, 178, 244, 307, 313 vector, 43, 85, 87, 88, 94, 97, 100, 179, 180, 181, 183, 185, 186, 187, 188, 190, 192, 193, 195, 197, 198, 199, 201, 203, 204, 205, 206, 207, 208, 212, 213, 214, 223, 224, 225, 228, 230, 233, 269, 292

Piezoceramic Materials and Devices, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Index velocity, 16, 22, 24, 38, 52, 56, 64, 116, 142, 148, 151, 165, 183, 193, 196, 197, 198, 212, 223, 226, 257, 293, 312, 313, 320 vibration, xii, 2, 3, 18, 64, 127, 142, 150, 152, 212, 213, 225, 285, 286, 287, 288, 289, 290, 300, 302, 308, 309, 310, 311, 313, 315, 316, 317, 318, 319, 320 vibrational modes, 125 viscosity, 201 visible, 18 visualization, 228 voids, 248, 274

W

X X-ray diffraction (XRD), 5, 12, 19, 24, 25, 33, 58, 61 X-ray diffraction data, 19

Y YBCO, 263, 266, 277 yield, 286

Z zinc, 114 zirconium, 268 Zn, 73

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warrants, 64 water, 24, 52, 123, 145, 194 water-soluble, 123 wave propagation, 138, 162, 163 wear, 116

335

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