UNIVERSITY OF CALGARY. Seyed Abdolali Zareian Jahromi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

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1 UNIVERSITY OF CALGARY Nonlinear Constitutive Modeling of Piezoelectric Materials by Seyed Abdolali Zareian Jahromi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING CALGARY, ALBERTA JANUARY, 2013 Seyed Abdolali Zareian Jahromi 2013

2 UNIVERSITY OF CALGARY FACULTY OF GRADUATE STUDIES The undersigned certify that they have read, and recommend to the Faculty of Graduate Studies for acceptance, a thesis entitled "Nonlinear Constitutive Modeling of Piezoelectric Materials" submitted by Seyed Abdolali Zareian Jahromi in partial fulfilment of the requirements of the degree of Doctor of Philosophy. Supervisor, Dr. Qiao Sun, Dept. Mechanical and Manufacturing Engineering Dr. Marcelo Epstein, Dept. Mechanical and Manufacturing Engineering Dr. Salvatore Federico, Dept. Mechanical and Manufacturing Engineering Dr. David Westwick, Dept. Electrical and Computer Engineering External Examiner, Dr. Christine Q. Wu, Dept. Mechanical and Manufacturing Engineering University of Manitoba Date ii

3 Abstract The well-known piezoelectric constitutive equations are applicable to piezoceramics driven by low to moderate fields and therefore representing linear behavior. However, these materials are also known for their ferroelectric and ferroelastic nonlinearities particularly when input fields exceed coercive strength. Such nonlinearities are caused mainly by micro-structural changes in the material known as domain switching. Therefore, a constitutive model that can accommodate the domain switching process can help capture the nonlinear behavior of the material. The past two decades have seen increasingly intensive activities in the attempt to build constitutive models to describe the piezoceramics nonlinearities. One of such models takes the advantage of existing linear constitutive equations. By decomposing the state variables into linear and internal components, the existing piezoelectric constitutive equations are augmented to include internal variables in describing the linear behavior while additional attention is devoted to dealing with the internal states pertinent to material's nonlinearities. The latter requires the modeling of the domain switching process. Together, the augmented linear constitutive equations and a domain switching model provide a complete nonlinear constitutive description. Although very successful in qualitatively capturing the polarization hysteresis and strain butterfly phenomena, a major shortcoming of this approach has been that it cannot match quantitatively with experimental data. iii

4 The objective of this research is to establish a constitutive model for piezoceramics that can not only qualitatively capture the material's nonlinear behavior over a large loading range, but also quantitatively match key characteristics from experimental data. In doing so, we have made three major contributions: 1) we discovered a flaw in the augmentation of linear constitutive equations for the use of materials in nonlinear regime and proposed a correct form of augmentation; 2) we proposed new domain switching rules based on the observations from single crystal experimental data; and 3) in the pursuit of model accuracy with quantitative description, we proposed the use of creep experiments to capture key characteristics related to timedependent and rate-dependent responses. A finite element approach is taken to carry out numerical modeling at the microscopic level. Experimental data available in literature have been used to help with model development and validation. iv

5 Acknowledgements I would like to acknowledge the contributions of those without which this work would not have been possible, and extend my sincere thanks to them. I am eternally grateful to my supervisor, Dr. Qiao Sun, whose encouragement, guidance and support enabled me to develop an understanding of my project. I acknowledge Dr. David Westwick for his time and effort in the revision of this thesis. To Ms. Ann Tikk and Ms. Nareeza Khan, whose work is greatly appreciated by the department s graduate students. Special thanks to my wife and family, who have provided me with continuous support and encouragement through the ups and downs of grad school. Last, and most importantly, I would like to thank my parents for all of their hard work, guidance, and support, and for the sacrifices they have made throughout my entire life. v

6 Dedication This thesis is dedicated to my mother and father, Who offered me unconditional love and support. Thank you for everything you have done for me! vi

7 Table of Contents APPROVAL PAGE... ii ABSTRACT...III ACKNOWLEDGEMENTS... V TABLE OF CONTENTS... VII LIST OF TABLES... X LIST OF FIGURES AND ILLUSTRATIONS... XI LIST OF SYMBOLS, ABBREVIATIONS AND NOMENCLATURE... XIII CHAPTER ONE: INTRODUCTION Piezoceramics in applications High-precision positioning Vibration damping Energy harvesting Nonlinearities in piezoceramics General nonlinear models Hysteresis models Creep models Piezoceramics nonlinear models Higher-order models Macroscopic models Microscopic models Phase field models Problem statement and anticipated contributions...19 CHAPTER TWO: CONSTITUTIVE RELATIONS OF PIEZOCERAMICS Background Definitions Piezoelectricity Elastic effect Dielectric effect Electrostrictive effect Polarizability and spontaneous polarization...24 vii

8 2.2.6 Pyroelectricity Ferroelectricity and ferroelasticity Linear constitutive relations Piezoceramics structure Nonlinearities and domain switching Hysteresis Creep Nonlinear constitutive relations Existing nonlinear relations Contradictory Cases Modified Equations More Case validation...42 CHAPTER THREE: MODEL DEVELOPMENT Switching model Experimental data in single crystals Experimental data of creep response Switching model The first model The second model The third model Finite Element model Governing equations Formulations Solution procedure Numerical simulation Material properties Grain orientations and polarization directions Piezoelectric specimen Procedure...70 CHAPTER FOUR: MODEL VALIDATION AND RESULTS Linear response Tuning switching models Single crystal response Polycrystalline response under pure electrical loading Cyclic command Old constitutive relations Rate-dependent response Constant loading Polycrystalline response under pure mechanical loading Cyclic loading Constant loading Polycrystalline response under combined loading Cyclic electric field with pre-stress Summary viii

9 CHAPTER FIVE: CONCLUSIONS Summary of results and contributions Future directions REFERENCES APPENDIX A: ELECTRIC DISPLACEMENT IN ONE-DIMENSIONAL PIEZOCERAMICS APPENDIX B.1: MATLAB CODES, FINITE ELEMENT MODEL APPENDIX B.2: MATLAB CODES, THE FIRST SWITCHING MODEL APPENDIX B.3: MATLAB CODES, THE SECOND SWITCHING MODEL APPENDIX B.4: MATLAB CODES, THE THIRD SWITCHING MODEL ix

10 List of Tables Table 2.1: Compressed matrix notation Table 3.1: Material properties of PIC151 [PI] Table 4.1: Parameters in switching models Table 4.2: Maximum and remanent values of response to a cyclic electric field Table 4.3: Maximum and remanent values of response related to different constitutive relations Table 4.4: Maximum and remanent values of longitudinal strain to a cyclic stress Table 4.5: Maximum and remanent values of longitudinal strain to constant stresses Table 4.6: Maximum values of electric displacement under different pre-stresses Table 4.7: Maximum values of longitudinal strain under different pre-stresses x

11 List of Figures and Illustrations Figure 1.1: Piezoceramics structure: domains and grains Figure 2.1: Unit cell of barium titanate above and below Curie temperature Figure 2.2: Six possible configurations of tetragonal unit cell in provskite crystals Figure 2.3: Hysteresis loop, electric displacemenent vs. electric field Figure 2.4: butterfly loop, strain vs. electric field Figure 2.5: One dimensional single crystal piezoceramics Figure 2.6: One dimensional piezoceramics with electrodes Figure 3.1: Six types of domains in tetragonal single crystals Figure 3.2: Switching process in single crystals Figure 3.3: Loading function, G 1, vs. dissipative energy associated with switching type α Figure 3.4: Saturation function, G 2, vs. volume fraction of a domain xi I ν Figure 3.5: Global and local coordinate frames Figure 3.6: The piezoelectric specimen and corresponding elements Figure 3.7: Flowchart of the applied nonlinear constitutive model Figure 4.1: Total displacement under a constant co-axial electric load Figure 4.2: Total displacement of the piezoelectric stack subjected to electric field by COMSOL Multiphysics Figure 4.3: The difference between the results from Finite Element and COMSOL Multiphysics Figure 4.4: Total displacement of the piezoelectric stack subjected to electric field by Finite Element model

12 Figure 4.5: Expermental data of Creep response, electric displacement vs. electric field [62] Figure 4.6: Electric displacement vs. time Figure 4.7: Longitudinal strain vs. time Figure 4.8: Displacement current density vs. time Figure 4.9: Electric displacement vs. cyclic electric field Figure 4.10: Longitudinal strain vs. cyclic electric field Figure 4.11: Transverse strain vs. cyclic electric field Figure 4.12: Comparison of the existing and the new constitutive relations Figure 4.13: Electric displacement vs. cyclic electric field with various rates Figure 4.14: Longitudinal strain vs. cyclic electric field with various rates Figure 4.15: Creep response, electric displacement vs. electric field Figure 4.16: Creep response, strain vs. constant electric field Figure 4.17: Creep response, electric displacement and strain, vs. electric field Figure 4.18: Longitudinal strain vs. cyclic compressive stress Figure 4.19: Transverse strain vs. cyclic compressive stress Figure 4.20: Longitudinal strain vs. compressive stress Figure 4.21: Transverse strain vs. compressive stress Figure 4.22: Electric displacement vs. cyclic electric field with pre-stress Figure 4.23: Dissipation energy density per electric field cycle vs. compressive prestress Figure 4.24: Longitudinal strain vs. cyclic electric field with pre-stress Figure 4.25: Transverse strain vs. cyclic electric field with pre-stress Figure A.1: A Gaussian surface in one-dimensional piezoceramics xii

13 List of Symbols, Abbreviations and Nomenclature Symbol b E C D E e α G G 1 G 2 G c K Description Body force per unit volume Elasticity constant at constant electric field Electric displacement Electric field Piezoelectricity constant Dissipative energy of switching type Loading function Saturation function Critical driving force Generalized stiffness K uu Mechanical stiffness matrix K, K Piezoelectric stiffness matrix uφ φu K Electrical stiffness matrix M m Generalized mass Mass matrix I n P P 0 s q v q R unit vector in the polarization direction of domain type I Polarization Magnitude of spontaneous polarization Free charge per unit area Free charge per unit volume Generalized force xiii

14 Rot T T c u X X, Y, Z x, yz, ε ε 0 I v σ α ω,, ε χ Rotation matrix Temperature Curie temperature Displacement Generalized displacement Global coordinate system Local coordinate system Strain Magnitude of spontaneous strain Volume fraction of domain I Mass density Stress Rate of switching of type Electrical potential Euler s angles Dielectric permittivity at constant strain Superscript I l s α Domain type Linear Spontaneous Switching type Abbreviation FE PID PZT SPM Finite Element Proportional, Integral and Differential Lead Zirconate Titanate Scanning Probe Microscope xiv

15 Chapter One: Introduction Piezoceramics have become very popular in smart structures due to their high efficiency of electro-mechanical energy conversion. Possessing this coupling capability, they can serve as sensors as well as actuators. Piezoceramics have been used in a wide range of applications such as high precision positioning, vibration damping, energy harvesting, strain gauges, etc. However, the applications of these materials are usually limited to their linear response due to inadequate knowledge about their nonlinear response. In particular, when used as high precision positioning devices, their performance quality in speed and range is limited due to the inherent nonlinear behavior. To fully achieve the potential of the materials and extend the range of applications, more accurate models that can describe the materials nonlinearities are desirable. In this introductory section, we describe some specific applications of piezoceramics and the errors caused by their inherent nonlinearities. In each application, the existing remedies to deal with nonlinearities are reviewed. In addition, the potential contributions that more accurate models can make to each application are discussed. Finally, we review the existing nonlinear models and introduce the main features of our model. 1

16 1.1 Piezoceramics in applications Although piezoceramics have been employed in a wide range of applications, they have much more potential that has not been used in order to avoid or eliminate nonlinear behaviors. We discuss some of these applications and their limitations High-precision positioning In the past three decades, nano- and micro- science and technology have emerged and explosively grown. Manipulation and interrogation at this scale necessitate positioning systems with high precision. The desired attributes of these systems are extremely high resolution, accuracy, stability, and fast response [1]. Piezoelectric actuators are usually employed in positioning systems, due to their high stiffness, compact size, and effectively high resolution [2]. Piezoelectric actuators are responsible to position a probe over a specific point or follow a trajectory with high precision. In scanning probe microscopy (SPM), for example, piezoceramics are used to position a probe over a sample with atomic scale resolution [3]. SPMs enable us to study and manipulate matters over the dimensions of several 100 m to 10 pm [4]. The invention of SPMs has fundamentally changed research in several areas, including biology, chemistry, material science, and physics [1]. However, there is an ever-present demand for SPMs that operate with higher performance [5]. The electromechanical performance of SPMs is analyzed with respect to several factors: precision level, scanning speed, and scanning range [6]. It is reported that the scanning speed is directly limited by the lowest structural resonance of the scanner [7]. However, increasing the scanning range can result in faster scanning of a surface. The precision level and 2

17 scanning range are limited by the intrinsic nonlinearities of piezoceramics [4, 8]. The effect of these nonlinearities on the scanner is explained in more detail in [9]. Hysteresis and creep are the major nonlinearities reported in the relation of the applied voltage and the resultant strain. They typically can cause a deviation as much as 25% in positioning of SPMs even within a small range of electrical loading [4]. These nonlinearities cause image distortion, because the scanner does not move linearly with the applied voltage, and the measurement points are not equally spaced. For example, the image of a surface with periodic structures will show non-uniform spacing and curvature of linear structures. Several methods have been proposed to compensate for the nonlinear behavior in piezoelectric actuators. One approach is to use displacement feedback of the actuator in a feedback control technique. Different types of feedback-based techniques such as PID, adaptive, and robust control are employed to mitigate the nonlinearities [10-11]. This approach is effective for low scan rates and has been used in some positioning systems successfully [4]. However, it suffers from several shortages. The main problem associated with employing feedback control in piezoelectric scanners is the noise in the sensor measuring displacement. Reducing signal-to-noise ratio of the sensor signal in high frequency loads would significantly limit the scanning speed of the device. Consequently, feedback has not been efficiently used in SPMs for precision positioning in horizontal directions [4]. In addition, displacement sensors with high precision are often expensive, and also their integration into the scanning stage of SPMs could be a difficult task due to space limitations [12]. Besides, some general limitations for feedback 3

18 control due to stability, input saturation, and dealing with non-minimum phase systems have been reported [4]. In another approach, it is proven that the use of charge or current control amplifiers can essentially reduce hysteresis [2, 13-16]. However, this operation can be prohibitively expensive as compared with the more commonly employed voltage control amplifiers. Another issue is that charge control is ineffective if maintaining DC offsets is desired, as required for numerous applications, e.g., the x-direction in an SPM must hold a specified position while a sweep is performed in the y-direction [17]. In addition, the higher output impedance of the current source restricts the slew rate. To obtain a high frequency response, piezoelectric actuators should be driven by voltage sources with low output impedance [12]. Feed-forward or model-based control scheme is an approach that does not require any additional sensors for implementation. Therefore, this method does not suffer from the issues associated with feedback techniques, e.g. noise and stability. In addition, in this method, voltage is employed to drive the actuator. Many researchers have been employing feed-forward or model-based control schemes [18-21]. A review of feedforward control approaches in accurate positioning can be found in [22]. The key idea of feed-forward control is to find inputs that achieve perfect tracking of the desired output. Consequently, a model of the system dynamics and an inversion method are the two fundamental parts of this method. The performance of feed-forward control schemes heavily relies on the accuracy of the system model. Therefore, model uncertainty is the main issue of employing this method. 4

19 In addition, combining feedback and feed-forward compensations has been shown to result in satisfactory tracking in piezoelectric actuators [23-24]. This combination illustrates better performance in the presence of plant uncertainties [25-26]. This method again suffers from the problems associated with the feedback method if uncertainties are too large. Therefore, a more accurate model can improve the performance of the control system. Moheimani [4] and Devasia et al. [1] presented a general survey of different types of control methods to compensate for the nonlinearities in piezoceramics Vibration damping Piezoceramics have been extensively employed in vibration damping applications due to their advantages such as high stiffness, high bandwidth, high efficiency, absence of moving parts and easy implementation [27]. For this purpose, piezoceramics are attached to the flexible structure in order to supply or absorb mechanical energy according to the designed control approach. One approach is passive control or piezoelectric shunt-damping [28-30], in which piezoceramics are shunted to an electric circuit. The piezoelectric transducers convert the mechanical energy of the structure into electrical energy, and then the electrical circuits dissipate the energy. In order to maximize the damping performance, both the converted energy by the piezoceramics and the dissipated energy by electrical systems should be maximized [31]. Different electrical networks have been developed to maximize the dissipative energy which are not the focus of this research. The converted energy depends on the piezoceramics constant, the vibration mode, and the position of piezoceramics within the structure [31]. Piezoceramics with high constant can increase the transferred energy. The vibration mode and location of piezoceramics depend on the design of the system. To analyze and 5

20 optimize a design, it is very beneficial to possess a dynamic model of the system, including both the piezoceramics and vibrating structures. A model that accurately operates over a wider range of loading can help to find a more realistic design. An alternative approach is active vibration control in which piezoceramics can serve as sensors, actuators, or both [27, 32]. When a structure vibrates, the piezoelectric sensors feed back the data to the control system which calculates the required load and then order the piezoelectric actuators to apply the appropriate load. A piece of piezoceramics can sense and apply force at the same time [27]. The performance of this approach depends on the control strategy as well as the design of the vibrating system including piezoceramics. Even in active control approach, which needs closed loop control, a model allows the designer to utilize model-based design tools and to simulate the characteristics of the system. In simple geometries, analytical methods, such as the assumed mode approach, have been used to analyze the vibrating structure [27]. For general geometries or boundary conditions, the Finite Element (FE) Method can be employed. Finite Element analysis of piezoelectric structures, either as sensors or actuators, has been performed too [33-34]. In these analyses, mostly, the response of piezoceramics is assumed to be linear. However, as mentioned in the previous section, piezoceramics can show nonlinear behavior and their properties change due to the loading history. Since in this application piezoceramics may operate under different electro-mechanical loading conditions, nonlinearities and changes can be more significant. Therefore, a more accurate model can help us design a more efficient vibration control system. 6

21 In addition, theoretically the area of the hysteresis loop represents the energy dissipated in each cycle [35-37]. This means that the hysteretic behavior of piezoceramics may also exhibit merits and can be employed to dissipate a portion of the mechanical energy. Several researchers studied the loss mechanisms in piezoceramics [36-40]. It has been shown that the loss in piezoceramics is significant especially under large loads and high frequencies [41]. Several factors contribute to the loss in piezoceramics. However, in the typical piezoelectric ceramic case, the loss is mainly caused by domain switching [42]. We have not seen any application of this potential in the published literature. Researchers mostly focus on the demerits of loss and the generated heat which can influence reliability and material properties [43] Energy harvesting The study and application of vibration energy harvesters are becoming more imperative due to the need of smaller and more efficient energy sources in advanced electronic technologies [44]. Vibration energy harvesting is the process in which the vibration energy of an ambient source is converted into electrical energy [45]. Piezoceramics, electromagnetic, and electrostatic converters are used in energy harvesters to perform this conversion. Among these converters, piezoceramics are very popular due to higher efficiency, higher output voltage, and no need of an external electrical source [46]. A wide variety of piezoelectric harvesting devices has been developed for different applications. A review of piezoelectric harvesters in civil applications is available in [47]. More review papers on energy harvesting can be found in [48-50]. Piezoceramics in harvesters act similarly to piezoelectric sensors: they convert mechanical into electrical energy, but in harvesters the amount of produced power is a 7

22 concern. Recent studies focus on optimizing the power generated from the mechanical energy [44]. To magnify the ambient vibrations, layers of piezoceramics are usually attached to a mechanical structure such as cantilever. These structures should be designed to operate around their resonant frequencies to generate more power. Another challenge is that the frequency of the vibration source may not be constant all the time. In this case, the device may be designed so that it can harvest the maximum power over the range of frequency. In addition to tuning the resonant frequency, their performance can be improved through modification of the piezoelectric materials, altering the electrode pattern, changing the poling and stress direction, layering the material to maximize the active volume, adding pre-stress to maximize the coupling and applied strain of the material [50]. To evaluate the design before production, we need accurate models that predict the behavior of the whole structure including piezoceramics nonlinearities. Similarly to the vibration control structures, different techniques such as analytical and numerical methods can be utilized to analyze the harvesting structure. Again, most of the existing papers consider linear models of piezoceramics. However, experimental studies illustrate the importance of nonlinearities in the performance of the harvester [45]. In addition to the nonlinear relation between mechanical displacement and output voltage, changes in the mechanical properties of piezoceramics such as stiffness and damping are very important in this application. These changes in material properties can be utilized to tune the resonant frequency of the device. Few studies consider nonlinear terms in stiffness or piezoelectric constants [44]. However, these studies did not consider the models proposed to predict the nonlinear response and changes in properties of piezoceramics. 8

23 1.2 Nonlinearities in piezoceramics Hysteresis and creep are two main nonlinearities associated with piezoelectric materials. Hysteresis means shortcoming, or delay, in the modern meaning, and originates from the Greek word ὑστέρησις. The electric displacement and strain response of piezoceramics to a periodic electrical or mechanical load follow closed loops known as hysteresis and butterfly loops, respectively. By definition, creep is a time-dependent response to a constant load over an extended period of time. In piezoceramics, the time-dependent behavior happens under electrical and mechanical loads, and at room temperature. These nonlinearities are defined and illustrated based on the macroscopic behavior of piezoelectric materials. However, they are caused by micromechanical changes, i.e. domain switching. In this section we briefly explain the origin of hysteresis and creep based on domain switching. We leave more details about the structure of piezoceramics and domain switching process to Chapter 2 of this thesis. Piezoceramics possess spontaneous strain and polarization due to asymmetry of atoms in their structure [51-53]. Spontaneous strain and polarization exist even in the absence of external loads. In a polycrystalline piezoceramic, a spatial region of a grain with the same direction of the spontaneous polarization forms a domain, as indicated in Figure 1.1.a. In an unpoled material, as depicted in Figure 1.1.a, the distribution of domains is random and therefore the overall spontaneous polarization is zero. When a mechanical or electrical load higher than a threshold level, called the coercive strength, is applied, the polarization of domains may be reoriented. For example, when an electric 9

24 field stronger than the coercive strength is applied, the polarization of the domains tends to align with the applied field, as shown in Figure 1.1.b. This process, known as domain switching, is irreversible by removing the external field. This switching is caused by changes in the position of ions in the crystal structure. Domain switching causes changes in overall spontaneous strain and polarization. These changes result in the nonlinearities such as electric displacement hysteresis and strain butterfly loops. Figure 1.1: Piezoceramics structure: domains and grains. a) In unpoled piezoceramics, domains are randomly polarized and the spontaneous polarization is zero; b) In poled piezoceramics, domain polarization aligns with the external field and the spontaneous polarization is not zero. Creep is the expression of the slow realignment of the crystal domains under a constant applied load, electrical or mechanical, over time [54]. When piezoceramics are subjected to electrical or mechanical loads, all domains try to align their polarization with the applied force. However, the switching does not happen at the same time. Some regions feel stronger loads due to local effects and switch faster. Some regions are subjected to smaller loads and consequently switch slower. 10

25 1.3 General nonlinear models One approach of modeling nonlinearities, such as hysteresis and creep, in piezoceramics is to employ general hysteresis and creep operators. These models mainly focus on the mathematical aspect of the problem and thus pay little attention to the physical mechanism. They predict the Hysteretic or creep relations between an applied load and the response of piezoceramics, e.g., the relation between applied electric field and strain. These models usually possess several mathematical parameters that are required to be adjusted using experimental data. Consequently, they may need a large amount experimental data for training. On the other hand, these models are usually simple, need less computational effort, and therefore are suitable for real-time applications. However, they are effective only for a narrow range of loading conditions. Some of the models commonly employed for predicting hysteresis and creep in piezoceramics are reviewed in this part Hysteresis models Hysteresis occurs in several physical phenomena such as plasticity, friction, ferromagnetism, ferroelectricity, ferroelasticity, etc. Hysteresis also appears in chemistry, biology, economics, even in experimental psychology, etc [55]. Although the reason for Hysteretic behavior in each field can be different, all hysteresis loops show some similarities. More details about hysteresis models are available in [55-57]. Several hysteresis models such as Prandtl-Ishlinskii, Krasnosel skii Pokrovkii, and Preisach have been employed in piezoceramics applications [58]. Among these models, the Preisach model and its modified versions are the most popular ones in piezoceramics applications. Preisach models originated for magnetic hysteresis, and have 11

26 subsequently been extended to piezoelectric materials [54]. The advantage of Preisach theory lies in its generality and strong mathematical foundations which provide a framework for quantifying hysteresis when the underlying physics is poorly understood. However, the generality of the technique also yields models which have a large number of nonphysical parameters. Hence, physical attributes of the data can be unclear, especially when identifying parameters or updating models to accommodate changing operating conditions is required. A number of extensions to the classical Preisach theory have recently been proposed to facilitate identification of parameters through correlation with physical principles [59]. Moreover, the original Preisach theory does not accommodate reversible effects and broadband operating conditions. The modifications required to accommodate these effects can significantly diminish the efficiency of resulting models [60] Creep models The creep phenomenon has been studied in other fields of material science such as metals. Although the mechanisms of creep may not be the same in all materials, there are mathematical similarities that can be employed for modeling. Macroscopic models of creep in piezoceramics mostly are based on the different power laws, following stress-relaxation models of creep in metals [61-62]. The kinetics of creep can be studied by measuring the relations between the strain rate, stress, and temperature. Stress-relaxation measurements are one way of obtaining this information [63]. 12

27 For characterizing the creep effect, rather than the logarithmic time displacement relationship, there are also models which can find their origin from modeling the viscoelastic behavior of creep by a group of springs and dampers [58]. 1.4 Piezoceramics nonlinear models According to the linear theory [64-65], four state variables are involved in the piezoceramics relations: mechanical stress, mechanical strain, electrical field, and electrical displacement. The actual data prove that the relations between these state variables are not actually linear, especially in high loading regimes [51]. The piezoceramics nonlinear models predict the nonlinear relations between these variables. Consequently, unlike general nonlinear models, these models can potentially predict the mechanical and electrical responses of piezoceramics under electro-mechanical loading. However, the accuracy of the predictions depends on the employed model. The existing models can be categorized into four different groups, which we review in this section. Higher-order and macroscopic models describe the whole polycrystalline as a single piece. Microscopic models define the response of each grain in the polycrystalline structure. Phase field models describe piezoceramics in sub-grain scale Higher-order models We note that the linear constitutive relations are derived from quadratic energy relations. A general approach to establish nonlinear constitutive relations is to consider higher order terms in the energy function [66-69]. Therefore, the resulting constitutive relations are nonlinear and possess some terms with the power of two or higher. Note that 13

28 these relations, similar to linear ones, are based on four variables: stress, strain, electrical field, and electrical displacement. The first and second order phase transition models belong to this category. We can achieve first and second order transitions by considering fourth and sixth order electric displacement terms in the energy relation [69]. In addition, the energy relation can be based on statistical mechanics yielding the Ising model. More detailed information about these models is available in [69]. These constitutive relations can predict instantaneous changes in electric displacement and show Hysteretic behavior. However, their non-unique relation and inadmissible results are sources of concern [70]. In addition, they cannot accommodate the ferroelastic aspect of response leading to strain-field butterfly loops [69] Macroscopic models The most appropriate approach for ferric materials is to consider internal variables related to the configuration of their structure [70]. In macroscopic models, these internal variables are related to the whole polycrystalline structure. Macroscopic models can describe the relations between the state variables and these internal variables using expressions induced by the behavior of system. For this reason, some researchers refer to them as phenomenological models. For these models, the computational effort is smaller than for microscopic ones. However, they are not as accurate as microscopic models in the sense that they only simulate some parts of the system. For many purposes in engineering it is simultaneously sufficient and efficient to model the macroscopic behavior directly. Several different models in this category have been introduced and employed to predict the nonlinear behavior of piezoceramics [71-80]. 14

29 Chen and his coworkers [70, 81-83] were one of the pioneer groups to apply the notions of material theory systematically to model the electromechanically coupled history dependence of piezoceramics. Chen et al. [70, 81-83] employed the degree of aligned dipoles in each direction as the internal variable. Their method could predict hysteresis and butterfly loops for a uniaxial electric field. The proposed set of equations is complex, even though it is not formulated for general electromechanical loadings. The next researchers approaching the nonlinear modeling of piezoceramics were Bassiouny and Maugin. They [84-87] employed plastic strain and residual polarization as internal variables. In [80, 88], the same variables are named remanent or spontaneous strain and polarization. They refer to the value of a state variable, strain or polarization, when load is removed and becomes zero [80, 88]. With this notion, state variables are decomposed into two components: one is reversible and linearly proportional to the magnitude of the input load and the other is spontaneous and irreversible caused by domain switching: ε ε ε l s ij ij ij D D P l s i i i (1.1) The idea of state variable decomposition has been adopted by many in an effort to construct constitutive relations that can be applied to material's nonlinear regime [80, 88-89] Lynch introduced a model with a broader range of validity [74]. The model is based on a set of internal variables characterizing the domain state in terms of the degree of relative spontaneous polarization and the degree of alignment to the applied load 15

30 direction. Defined by the additive decomposition, Eq. ( 1.1), spontaneous polarization and strain are given as functions of the internal variables. The model is capable of representing the dielectric butterfly, and ferroelastic hysteresis. An interesting phenomenological constitutive law for ferroelastic switching in piezoceramics was introduced by Landis in 1999 [90]. The basis of this purely mechanical model is the additive decomposition Eq. ( 1.1). As an interesting aspect, the model takes into account the dependence of the anisotropy of the elastic properties on the spontaneous strain in terms of the principal values of this tensor. The evolution equation for the ferroelastic strain is given by a flow rule just as in incremental plasticity. As a result, the uniaxial ferroelastic hysteresis is represented properly in the loading direction as well as in the direction transverse to loading. In 2002, Landis improved his macroscopic model for the fully coupled electromechanical constitutive behavior of piezoceramics [78]. As it seems, the complete range of ferroelectric and ferroelastic hysteresis phenomena is covered by this thermodynamically consistent model by means of a sophisticated switching surface depending on appropriately chosen invariants. Kamlah presented a complete literature review of the models proposed to predict elecro-mechanical hysteresis phenomena [53]. In addition, he proposed a macroscopic model describing the main large signal hysteresis phenomena in piezoceramics. The model was developed for the purpose of structural mechanical analysis by means of the Finite Element Method Microscopic models We now turn towards the discussion of microscopic models relying on advanced methods of continuum mechanics and micromechanics. It is known that the peculiar 16

31 microstructure of piezoceramics determines their nonlinear macroscopic response [51]. Therefore, several efforts have dealt with the computation of macroscopic properties starting from microscopic models in order to achieve a more quantitative understanding of the nonlinearities observed in piezoceramics. These models describe the response of piezoceramics at the grain scale. The same additive decomposition, Eq. ( 1.1), is usually considered in microscopic models too. Many of these models assume the polycrystalline structure to consist of monodomain grains, the polarization of which switches by a discrete angle if an energetic switching criterion is met [88, 91-92]. Instead of simply averaging the fields over uncoupled grains for the computation of the macroscopic response [88, 92], some researchers have taken the interaction among the grains into account by Eshelby inclusion methods [91] or explicitly by the Finite Element Method [80, 93]. Early works in this category have treated the single crystal behavior in an over simplified manner by assuming domain switching is a sudden change from one variant to another [88, 92]. A significant improvement has been made by Huber et al. [94] who introduced the incremental switching analogous to the operation of a slip system in crystal plasticity. The kinematics of the process due to switching types is developed in analogy to crystal plasticity. Driving forces for the transformation are derived from the excess of the external work rate over the dissipation due to domain wall motion and a switching type happen if the driving force reaches a critical value. In this way, the existence of the domain structure and its influence on the grain response are taken into account. From this single crystal model, the macroscopic response of the polycrystalline is derived 17

32 approximately by a self-consistent scheme. Experimental verifications of this model have been provided by [52]. In 2005, Kamlah and his coworkers modified Huber s model to introduce rate dependence [80]. To compute the constitutive behavior of ferroelectric ceramics, they used a plane strain finite element model, where each element represents a single grain in the polycrystalline. They employed the Finite Element formulation for electromechanical boundary value problems developed by Landis [95]. Seelecke and his coworkers [96-98] have proposed different frameworks for switching in ferroelectric single crystals. They generalize the one-dimensional free energy model by Smith et al. [99] to two- and three-dimensional cases based on the theory of statistical thermodynamics Phase field models Recently, phase field models have also been employed to study the polarization switching and nonlinear behavior of ferroelectrics [100]. These models are very powerful in predicting the domain boundaries in a grain. However, their application to polycrystalline piezoceramics is not time efficient. In phase field models, the polarization is considered as a parameter that can change its orientation and magnitude under the external mechanical and electrical loadings. The total free energy in phase field models includes the Landau free energy, which is expressed as a polynomial in the polarization [100]. The space discretization in a phase field model can be implemented by the Finite Difference Method or the Finite Element Method. For example, Schrade and his coworkers [ ] and Su et al. [103] conducted two-dimensional Finite Element simulations on ferroelectrics based on the 18

33 Landau free energy. Wang et al. [100] developed a three-dimensional nonlinear Finite Element formulation for the static problem of ferroelectric materials without temporal evolution using the Landau free energy and employed it to investigate the polarization distribution near a notch in a ferroelectric single domain subjected to mechanical and electrical loadings [100]. 1.5 Problem statement and anticipated contributions The existing models for piezoceramics can qualitatively predict the response of polycrystalline piezoceramics. However, they suffer from several limitations in predicting all aspects of the piezoceramics response to general electro-mechanical loads. In this thesis, we propose a new constitutive model in an attempt to improve the accuracy of the predicted response. In particular, we propose a set of constitutive relations and three rate-dependent switching models, resulting in a model that can be placed in the category of microscopic models. A three dimensional Finite Element model is employed to apply the constitutive model. The additive decomposition in Eq. ( 1.1) is usually combined with the linear constitutive relations in microscopic nonlinear models. However, in our recent studies, we found inconsistent results in simple thought experiments with the current augmented constitutive relations. In this thesis, we present these cases and upon analyzing the results, we propose a correct version of the augmented constitutive equations. Most researchers do not conduct validation tests for their switching models, but just compare the results of their complete constitutive models for polycrystalline piezoceramics with experimental data. However, there is not much effort to validate the 19

34 switching model against the experimental data from single crystalline piezoceramics. Here, we focus on single crystals in order to understand the mechanism of the switching process without the complication of grain orientation and inter-grain interactions brought forward by polycrystalline ceramics. We propose new switching models developed based on the main features of switching in single crystals. In addition, we propose the use of creep experiments to capture key characteristics related to time-dependent and ratedependent responses in the pursuit of model accuracy with quantitative description. Each switching model contains several parameters that are required to be adjusted based on the piezoelectric material. We employ the experimental data of the material s creep response under constant electric field to justify these parameters. The anticipated contribution and methodology of this thesis can be summarized as follows: Proposal of constitutive relations for piezoceramics which can include the spontaneous terms properly. Proposal of a suitable function for the rate of switching. This function should agree with time and rate dependent response of piezoceramics reported in the literature. Therefore, we should first understand the origin of this response by focusing on the switching in single crystals. In addition, the creep response can reveal more information about the characteristics related to time-dependent and rate-dependent responses. Development of a Finite Element model employing the nonlinear constitutive law for a polycrystalline material. 20

35 Prediction of the response of piezoceramics subjected to different combination of electromechanical loading and validation the results with reported data. The remainder of this thesis is organized as follows. In Chapter 2, we briefly review the fundamental concepts and linear constitutive relations of piezoceramics. Then, we turn into the material structures and explain how the changes in their micro-structure result in hysteresis and creep. In addition, we review the existing nonlinear constitutive relations and introduce our set of equations. Chapter 3 focuses on developing different parts of our constitutive model, and introduces our proposed switching models, based on the switching in single crystals. Then, the Finite Element model is developed based on the proposed constitutive relations. Finally, the numerical procedure we have employed to analyze the behavior of piezoceramics, is explained. In Chapter 4, we present the results of our constitutive model and compare them with experimental data. We investigate the performance of our model in three different cases. The first case is related to the linear response of piezoceramics. In second case, the response of single crystal material is studied. The third case is related to the response of polycrystalline material under different mechanical and electrical loading conditions. Finally, in Chapter 5, we summarize the work presented in this thesis and outline the possible future extensions of this research. 21

36 Chapter Two: Constitutive Relations of Piezoceramics In this chapter, the fundamental concepts and linear constitutive relations of piezoelectric materials are briefly reviewed. Then, we focus on their crystal structure and explain their nonlinearities, based on the changes happening in the structure, and domain switching. In addition, we review the existing nonlinear constitutive relations and present the related inconsistent results in simple thought experiments. Finally, we propose a set of new relations, which we then test in our thought experiments. 2.1 Background In 1880, Pierre and Jacques Curie discovered that in natural crystals, such as quartz, tourmaline, and Rochelle salt, pressure can generate electric charge. In 1881, the term "piezoelectricity" was first suggested by W. Hankel, and the inverse effect was deduced by Lipmann from thermodynamics principles [64]. In the following three decades, collaborations within the European scientific community established the field of piezoelectricity, and by 1910, Voigt published a standard reference work detailing the complex mechanical and electrical relationships in piezoelectric crystals [104]. However, the complexity of the science of piezoelectricity made it difficult to mature until a century later when the microstructure of piezoelectric material was studied [64]. 22

37 2.2 Definitions Before focusing on the crystal structure of piezoceramics and discussing their constitutive relations, we review the definition of the related concepts Piezoelectricity Piezoelectricity stems from the Greek word piezo for pressure and thus means electricity resulting from pressure. Cady [64] has defined piezoelectricity as the electric polarization produced in certain materials in response to the applied mechanical stress. In the linear materials, the electric polarization is proportional to the stress and its direction reverses if the sign of stress changes. This is called direct piezoelectric effect, and always accompanied to the inverse piezoelectric effect which is referred to the mechanical deformation caused by the applied electric field in these materials [64]. The piezoelectric effect is understood as the linear and reversible interaction between mechanical and electrical states [69]. The piezoelectric effect can be found in different materials such as certain ceramics, polymers, and biological materials. The focus of this research is on the piezoelectric ceramics or piezoceramics Elastic effect Elastic effect refers to the reversible interaction between stress and strain in mechanical state. For small strains, most elastic materials, exhibit linear elasticity. This means that the relation between stress and strain is linear. In three dimensions, the fourth order elasticity tensor, the components of which are the so-called elastic constants of the material, is required to relate stress and strain. 23

38 2.2.3 Dielectric effect By definition, a dielectric material is an electrical insulator that can be polarized in response to the applied electric field. When a dielectric material is subjected to an electric field, electric charges slightly move from their equilibrium positions causing electrical polarization. The displacement of charges is such that the resultant electric field opposes the external field and thus reduces the overall field within the dielectric material. In general, a second-order tensor, the susceptibility tensor, is needed to define the relation between electric field and dielectric polarization. The electric susceptibility is a measure of how easily a dielectric polarizes under an electric field Electrostrictive effect When a material of any kind is subjected to an electric field, movement of charges also causes strain. This property is known as electrostriction. The resulting strain is proportional to the square of the resulting polarization. In piezoceramics, piezoelectricity dominates over the electrostrictive effect [53] Polarizability and spontaneous polarization As mentioned in 2.2.3, dielectric materials show electrical polarization under an external electric field. These materials that can show polarization in response to an external load are polarisable. Some dielectric materials show polarization even when no external load is applied. This is spontaneous polarization and is caused by asymmetry of charges in the structure of the material Pyroelectricity The value of spontaneous polarization is related to the temperature. This property is pyroelectricity and is utilized in sensing the temperature. Piezoceramics are always 24

39 pyroelectric. However, since our focus is on the electro-mechanical interaction of piezoceramics, we assume all processes to be isothermal. Thus, pyroelectricy is out of scope of this thesis Ferroelectricity and ferroelasticity In some materials, spontaneous polarization can be reoriented in response to a strong applied electric field. This property is ferroelectricity and these materials are known as ferroelectric materials. This phenomenon is the main source of hysteresis in piezoceramics. Ferroelasticity is the mechanical equivalent of ferroelectricity. When a strong stress is applied to a ferroelastic material, it can exhibit reorientation of spontaneous strain. 2.3 Linear constitutive relations In this section, we briefly review the linear constitutive equations of piezoelectric materials. Since temperature dependent behavior of piezoceramics is out of scope of this work, we neglect the thermal effects in the equations. The relations for the reversible linear behavior of piezoceramics are well developed through thermodynamic approaches and have been utilized in some applications [64-65, 104]. The linear constitutive equations for piezoceramics can be expressed as following: C e E, and (2.1a) ij E ijkl kl kij k D e E. (2.1b) i ijk jk ij j 25

40 where, E C ijkl, e ijk, and ij represent the elasticity tensor, piezoelectricity tensor, and dielectric permittivity tensor, respectively. The superscripts of elasticity and dielectric tensors denote that these properties are considered at constant electric field and strain. These equations describe the stress ij and electric displacement strain ij and electric field D i as a function of E i. There are alternative sets of constitutive relations with different sets of dependent and independent variables [69]. Which set should be used depends on the application and the circumstances. These different sets of constitutive equations are all exact and can be converted into each other. In this work, we prefer the relations of stress and electric displacement, which are appropriate for Finite Element analysis. In order to write elastic and piezoelectric tensors in matrix form, the Piola-Voigt compact matrix notation can be employed [64]. This compact notation replaces the subscripts of ij and kl by p and q, according to Table 2.1. As shown in Table 2.1, i, j, k, and l take the values from 1 to 3; but p and q take the values from 1 to 6. Table 2.1: Compressed matrix notation ij and kl p and q or or or

41 Then the constitutive equations of ( 2.1) can be expressed as: C e E, and (2.2a) p E pq q kp k D i e E. (2.2b) iq q ij j 2.4 Piezoceramics structure Piezoceramics are of crystalline nature and thus their lattice consists of unit cells, whose repetition in space produces a crystal lattice. The unit cell is the smallest divisible unit of a crystal that possesses the symmetry and properties of the crystal. The unit cells of piezoceramics are not symmetric and therefore possess spontaneous polarization [51-53, 105]. The spatial region of neighboring unit cells possessing parallel polarization makes a domain, and a group of domains with the same crystal orientation makes a grain. Some piezoceramics are single crystals, meaning that the whole piece consists of only one grain. For example, piezoelectric effect was first discovered in single-crystal quartz. The most widely used piezoceramics, such as barium titanate (BaTiO3) and lead zirconate titanate (PZT), consist of multiple grains and are thus known as polycrystalline. Normally, polycrystalline piezoceramics are easier to fabricate and more durable that single crystals [106]. 27

42 Figure 2.1: Unit cell of barium titanate above and below Curie temperature Above Curie temperature the unit cell is cubic, but below that temperature the unit cell is not cubic and possesses spontaneous polarization and strain. In ferroelectric piezoceramics, unit cells switch their polarization direction when subjected to sufficiently strong mechanical and electrical loads. This process is called domain switching and happens to achieve a more stable state with less energy. In this process, the domains with high energy shrink and the ones possessing lower energy grow. Consequently, the average properties of the grain will change causing a nonlinear response. Barium titanate and PZT have perovskite-type structures. Above a critical temperature T c, known as the Curie temperature, perovskite materials possess paraelecric state, i.e. non-ferroelectric state, and their unit cell is cubic. Below T c, the unit cell is non-cubic and the material is piezoelectric, ferroelectric, and ferroelastic. Figure 1.1: shows a schematic view of the tetragonal unit cell of barium titanate including the ions. Although other crystallographic structures, such as rhombohedral, do arise [36]; in this thesis, we focus on materials that are tetragonal in the ferroelectric state. 28

43 As depicted in Figure 1.1:, in tetragonal form of provskite crystals, the central ion shifts toward one face. Consequently, the mean centers of opposite charges do not coincide and a dipole moment of the charge distribution exists. Thus, the unit cell possesses a spontaneous polarization. On the other hand, the shift of the central ion elongates the unit cell in one direction and shortens in the other two directions. Thus, it causes a spontaneous strain aligned with the dipole moment. The central ion can move toward every face-centered negative charges. Therefore, there are 6 possible directions to shift in a three dimensional unit cell. Figure 2.2 illustrates the possible configurations of a tetragonal unit cell. Consequently, 6 possible domains can be formed in each grain. When a piece of piezoceramics cools down from the Curie temperature, the number of unit cells in each configuration is the same. This state is known as virgin state. In the virgin state, the total polarization is zero and the whole piece does not show piezoelectric effects. To illustrate piezoelectric effects, this piece should be polarized. This can be done by applying a strong electric field so that polarization in all unit cells aligns to the direction of field. 29

44 Figure 2.2: Six possible configurations of tetragonal unit cell in provskite crystals The six possible configurations of tetragonal unit cell can switch to each other if subjected to sufficient electrical or mechanical load. This switching process, domain switching, is irreversible by removing the load and causes the nonlinear behavior of piezoceramics. On the other hand, the reversible portion of the response is linearly proportional to the amount of applied loads without reorientation of domains. 2.5 Nonlinearities and domain switching In this section, we explain the macroscopic nonlinear behavior of piezoceramics based on the microscopic switching happening in their structure. In this research, we focus on the rate of switching in order to predict the rate-dependent behavior of piezoceramics. 30

45 2.5.1 Hysteresis As mentioned in the first chapter, the electric displacement of piezoceramics follows a hysteresis loop when subjected to a cyclic electric field. Under this cyclic loading condition, the strain follows a butterfly loop. Figure 2.3 illustrates a schematic view of the hysteresis loop, and 2.4 shows the butterfly loop. These curves are the response to a triangular electric field: the electric field increases linearly from zero to a maximum value Em, then reduces to Em, and finally returns to zero. As Figure 2.3 and 2.4 show, at point 1, the material is at virgin state and therefore electric displacement and strain are zero. When the electric field is small, only a small number of cells switch due to intra-granular forces. Therefore, the linear piezoelectric effect is dominant and the material shows linear behavior. At coercive electric field, point 2, domain switching speeds up and the polarization and strain rapidly increases. This is the result of the realignment of unit cells. The switching process continues until almost all the cells switch, point 3. This phenomenon is known as domain saturation or exhaustion. After saturation, again linear response becomes dominant since no more cells are available to switch. It is noticeable that the slope of strain and polarization at saturation is bigger than the slope at the initial portion of the response. This is because the material is poled at saturation and thus possesses piezoelectric properties while in the initial portion of response the material is not piezoelectric. While electric field starts reducing, from point 4, the cells keep their situation and no switching happens. Consequently, the electric displacement and strain both reduce linearly. However, when the electric field reduces more, near point 5, the internal residual stresses make some cells switch resulting in small nonlinearities. When the electric field 31

46 returns to zero, point 5, the material still presents electric displacement and strain. These values are remanent polarization and strain and are the results of alignment of unit cells in one direction. While the electric field reduces more, the switching rate increases again but in reverse direction, point 6. As a result, the electric displacement and strain reduce rapidly up to the point 7. At point 7, the orientations of unit cells are such that the total electric displacement is zero. However, the total configuration is different from point 1 at which strain was zero. By reducing the electric field more, more unit cells switch their polarization to the reverse direction and therefore the electric displacement vector reverses and becomes negative. However, strain increases due to fact that the elongation of unit cells with spontaneous polarization upward and downward does not change. The strain increases until all cells align in the downward direction, while electric displacement reduces. Again by increasing electric field, the same scenario happens but in the opposite direction. The electric field at point at which the polarization become zero is defined as the coercive field. 32

47 Electric Displacement Em 0 Em Electric Field Figure 2.3: Hysteresis loop, electric displacemenent vs. electric field The electric displacement response to a triangular electric field that increases linearly from zero to a maximum value Em, then reduces to Em, and finally returns to zero in each cycle. 4 5 Strain Em 0 Em Electric Field Figure 2.4: butterfly loop, strain vs. electric field The strain response to a triangular electric field that increases linearly from zero to a maximum value Em, then reduces to Em, and finally returns to zero in each cycle. 33

48 2.5.2 Creep Creep behavior is caused by the slow realignment of the unit cells. When a ferroelectric ceramic subjected to an electrical or mechanical load, all domains try to align their polarization with the applied force. The unit cells located at the boundary of domains, domain wall, and some specific unit cells inside the domains which are near to crystal defects have more potential to switch. The switching happening at the boundary is known as domain wall motion and the latter one is nucleation. Nucleation happens in a very short time while the propagation of a domain by domain wall motion takes time. Contributions of domain switching to the creep behavior depend on the magnitude of the load applied and the size of domains available for switching. For an initially unpoled specimen, electrical creep and compressive creep first increase with increasing the applied electric field and stress. At load levels near the coercive field or coercive stress, creep is maximum due to the highest amount of domain switching. Then, a gradual exhaustion of the reservoir of switchable domains causes a decay of the creep effects with increasing load. It is confirmed that the creep of piezoceramics is of primary or transient type which is characterised by a continuous decrease of the creep rate over time. This result indicates that the creep mechanism is saturated or exhausted. In other words, the total amount of domain switching, which can be induced under a constant load, is limited [62]. 2.6 Nonlinear constitutive relations The linear constitutive relations introduced in 2.3 cannot accommodate ferroelectric/ferroelastic changes such as hysteresis and butterfly loops [51-53]. 34

49 2.6.1 Existing nonlinear relations As briefly reviewed in the first chapter, a general approach to establish nonlinear constitutive relations is to consider higher order terms in the energy function [66-69]. For example, the Helmholtz energy relation for the second order phase transition can include quadratic and quartic polarization terms and linear and quadratic strain terms [69]. As illustrated in [69], these higher order terms enable the characterization of hysteresis. However, their non-unique relation and inadmissible results are sources of concern [70]. In addition, they cannot accommodate the ferroelastic aspect of response leading to strain-field butterfly loops [69]. An alternative approach is to consider internal variables related to the configuration of domains [70]. Chen et al. [70, 81-83] employed the degree of aligned dipoles as the internal variable. Their method could predict hysteresis and butterfly loops for materials subjected to one dimensional electric field. Bassiouny and Maugin [84-87] employed plastic strain and residual polarization as internal variables. In [80, 88, 107], the same variables are named remanent or spontaneous strain and polarization. They refer to the value of a state variable, strain or polarization, when load is removed and becomes zero [80, 88, 107]. With this notion, the state variables are decomposed into two components: one is reversible and linearly proportional to the magnitude of the input load and the other is spontaneous and irreversible caused by domain switching: ε ε ε, and (2.3a) l s ij ij ij D D P. (2.3b) l s i i i 35

50 This decomposition concept agrees with the response of piezoceramics explained in the last section. This idea of state variable decomposition, Eq. ( 2.3), has been adopted by many in an effort to construct constitutive relations that can be applied to the nonlinear regime [80, 88, 107]. Recognizing that the constitutive nonlinearity is mainly caused by domain switching, which is reflected in the internal state variables, namely, spontaneous strain and polarization, Hwang et al. [88] combined Eq. ( 2.3) with the linear constitutive relations, Eq. ( 2.1), to yield: E s σ C ( ε ε ) e E, and (2.4a) ij ijkl kl kl kij k D P s e ( ε ε s ) χ ε E. (2.4b) i i ijk jk jk ij j Although simple and straightforward, the above equations deserve a few remarks. It describes a general relationship of the material's response to excitation inputs at all s s levels. Besides the extra terms of internal variables ( ε, P ), Eq. ( 2.4) differs from the linear constrictive relation, Eq. ( 2.1), in that the material properties in Eq. ( 2.4) are no longer constant, but are related to the material's domain configuration. In other words, they are dependent on the internal variables of spontaneous strain and polarization. The significance of Eq. ( 2.4) is that it provides a relationship between the material states for s s given internal states ( ε, P ). As such, a complete nonlinear constitutive law requires ij i another portion which can derive the internal states. For example, Huber et al. [107] introduced a microscopic model that can describe the domain switching process and leads to the determination of spontaneous strain and polarization as well as material property tensors. This approach has been followed by many researchers [80, ]. Published 36 ij i

51 results show that it can qualitatively generate hysteretic polarization-field and butterfly strain-field behaviors. In the following sections, we focus on examining the validity of Eq. ( 2.4). Although seemingly a straightforward augmentation from the linear constitutive equation ( 2.1), Eq. ( 2.4) provides doubtful conclusions in the thought experiments to be described in the following sections. The question is whether the state variable decomposition Eq. ( 2.3) is proper reflecting the effect of spontaneous polarization on both electric field and electric displacement variables. Although the state variable decomposition seems to be intuitive, the electric field variable is also affected by the spontaneous polarization. The electric variables, namely electric field, electric displacement, and polarization are not independent and hence extra care needs to be taken when interpreting the effect of spontaneous polarization. In the following sections, we describe two simple thought experiment cases and their results by using Eq. ( 2.4) and by direct physical interpretation. Upon analyzing the cause of inconsistent results, we propose the correct augmentation of Eq. ( 2.1) for the general constitutive description of the piezoceramics material under all load levels. In addition, we test the proposed equations with two more cases to validate its correctness Contradictory Cases We consider a hypothetical one-dimensional problem where a single crystal piezoelectric material is placed in vacuum, free of external loads, mechanical or electrical. The material is homogenous and in slab shape having finite width but infinite depth and length to avoid boundary effects and justify the use of a one-dimensional problem. We first consider a case where all dipole moments are aligned in one direction 37

52 to give a uniform spontaneous polarization field s P in the same direction as the dipole moments, as shown in Figure 2.5.a. The material also possesses a uniform spontaneous strain s ε field in its width direction. Since the problem is one-dimensional, the tensors describing the material properties can be reduced to scalar quantities. We therefore drop the indices in Eq. ( 2.4) for simplicity. The material is under zero stress, that is, σ 0, because no external loads or constraints exist in the plane of interest. There are no electrodes on the surface of the material and therefore no free charges. The material has zero electric displacement, D=0 (See Appendix A for a proof). Setting σ and D to zero, Eq. ( 2.4) determines the effective electric field as a consequence of the spontaneous polarization 1 1 ( T s E ec e χ) P (2.5) This is the depolarization field which is in the opposite direction of the spontaneous polarization field. The question is whether this depolarization field should cause additional strain beyond the initial state of strain: the spontaneous strain. By definition, the spontaneous strain is the total strain of the material that is irreversible when the external field is zero. We expect at this point that the total strain ε is equal to the spontaneous strain However, solving Eq. ( 2.4) for strain yields: s ε since nothing is done to change the initial state of the material. ε ε C e ( ec e χ) P s 1 T 1 T 1 s (2.6) In Eq. ( 2.6), 1 C is elastic compliance and ( 1 T 1 ec e ) is the impermittivity component. According to the properties of poled piezoceramics, the second term on the right hand side of Eq. ( 2.6) is non-zero. Obviously, the strain is different from the initial 38

53 state s ε even though nothing physical has been done to cause such change. The existing nonlinear constitutive equation ( 2.4) provides inconsistent results with physical interpretation. Figure 2.5: One dimensional single crystal piezoceramics a) Left: dipoles of all unit cells are aligned in the same direction; Right: consequent fields of spontaneous polarization and strain; b) Left: dipole moments of unit cells are equally distributed in the upward and downward directions; Right: consequent fields of spontaneous polarization and strain. We consider another case as shown in Figure 2.5.b. It is similar to the previous case except for the arrangement of dipole moments. In this case, half of the cells have dipole moments pointing upward and the rest point downward. Hence, the spontaneous s polarization field is zero due to the volumetric averaging of the dipole moments: P 0. However, the material possesses the same amount of spontaneous strain ε s as in the previous case. This can be explained by the fact that dipoles are caused by the shift of ion 39

54 positions. When ions move to opposite positions, the direction of the dipole moment is reversed but the material strain remains the same. Therefore, we expect the constitutive equation, Eq. ( 2.4), to indicate the same total strain as Eq. ( 2.6). Again, stress and electric displacement are both zero for the same reason as in the previous case. With s σ 0, P 0 and D 0, Eq. ( 2.4) states that there will be no electric field inside the material, and the strain is E 0, ( 2.7) ε s ε. ( 2.8) Obviously, the constitutive equation ( 2.4) provides a different strain even though the two cases have physically identical strains. We therefore question the validity of Eq. ( 2.4). From results given in Eq. ( 2.6) and ( 2.8), it is clear that the difference is caused by the depolarization field that contributes to the effective electric field which in turn determines the states Modified Equations The derivation of Eq. ( 2.4) is based on the fact that material nonlinearity is measured in the response of state variables to external loads. Thus the decomposition of state variables, ( ε, D ), is employed to result in the form of Eq. ( 2.3). However, it is important to note that although observation of electric displacement D is usually convenient, it is the behavior of dipole moments and hence P s i that is causing the nonlinearity in material's response to external loads. A more direct effect of 40 s P i is on the effective electric field, which in turn influences the state variables, both strain and the

55 electric displacement. In other words, when augmenting the linear constitutive equations to accommodate materials nonlinearity, variable decomposition should be done to the electric field rather than electric displacement. We propose the following decomposition: ε ε ε, and (2.9a) l s ij ij ij 1 E E χ P. (2.9b) l ε s i i ij j Note that the difference between the above decomposition and Eq. ( 2.3) is in the electric variables. Equation ( 2.9b) signifies the effect of spontaneous polarization on the electric field which in turn influences both electric and mechanical states, ε and D. On the contrary, state decomposition as done in Eq. ( 2.3), when applied to the linear constitutive equations, Eq. ( 2.4), does not account for the effect of spontaneous polarization on strain. Decomposition ( 2.9b) implies that the reversible or linear components in the state variables are driven by the field strength E l i rather than E i. Consequently, Eq. ( 2.9b) is used to replace the respective variables in the linear constitutive relations of eq. ( 2.1) to yield: 1 E s ε s σ C ( ε ε ) e E e χ P, and ( 2.10a) ij ijkl kl kl kij k kij kl l D P s e ( ε ε s ) χ ε E. (2.10b) i i ijk jk jk ij j ε Compared with Eq. ( 2.4), the above equation has an extra term, 1 e χ P the strain expression. This extra term cancels out the effect of depolarization field on the strain. On the other hand, the electric displacement field equation ( 2.10b) is identical to the existing expression eq. ( 2.4b). kij kl s l, in 41

56 To validate the modified constitutive equations, we apply Eq. ( 2.10) to the two cases we used in the last section. For the case shown in Figure 2.5a, the electric field and the total strain are: E χ P 1 s, and ( 2.11a) For the case shown in Figure 2.5b the results are ε s ε. (2.11b) E 0, and ( 2.12a) ε s ε. (2.12b) We now have consistent results of strain for both cases. The difference is the electric field because of the different spontaneous polarization field in each case. In the next section, we further examine the modified constitutive equation ( 2.10) with more case studies More Case validation In this section, we present two more hypothetical one-dimension cases as shown in Figure 2.6. In these cases, a slab of piezoceramic material possessing uniform polarization in its width direction is placed between two parallel conducting plates. The spontaneous strain and polarization are ε s and are reduced to scalar quantities. s P. Again, all material property tensors 42

57 Figure 2.6: One dimensional piezoceramics with electrodes. a) Parallel plates are disconnected producing zero external electric field. b) Parallel plates are connected to create an external electric field to result in zero field strength inside the material In Figure 2.6.a, the two conducting plates are not connected. Hence, the electric field external to the piezoceramic material is zero. The effective field at any point inside the material is the depolarization field caused by the spontaneous polarization: 1 s E χ P. To quantify the state variables, we use the proposed constitutive equation ( 2.10). This is again a case with no stress, σ 0, because the material is free of mechanical loads and constraints in the plane of interest, With σ 0 and E 1 s χ P, Eq. ( 2.10) concludes D 0, and ( 2.13a) ε s ε. (2.13b) This makes sense because, similarly to the previous cases, no external loads are applied to alter the material's states. Strain remains the same as the material's spontaneous 43

58 strain. The electric displacement is zero because charges on the plates cannot move from one to another. On the contrary, Eq. ( 2.4) once again gives erroneous results in both electric field and strain: 2 1 s D e ( Cχ) P, and ( 2.14a) s 1 s ε ε ecχ ( ) P. (2.14b) Finally, we consider a case where the two conducting plates are connected as shown in Figure 2.6.b. Free charges on the plates move to neutralize while leaving those of opposite polarity to the medium on the plates, thereby creating an external field whose strength is equal to the spontaneous polarization field. Thus, the electric field inside the piezoelectric material must be zero: E 0. Since the material is free of both mechanical load and constraints in the plane of consideration, the stress is again zero. We use both sets of constitutive equations to see what the state variables would be. With σ 0 and E 0, Eq. ( 2.10) concludes 2 1 s D (1 e ( Cχ) ) P, and ( 2.15a) s 1 s ε ε ecχ ( ) P. (2.15b) However, results obtained by means of Eq. ( 2.4) are different: D s P, and ( 2.16a) ε s ε. (2.16b) In this case, the non-zero external electric field must have the effect of causing deformation by the very nature of the piezoelectric materials. Strain in Eq. ( 2.16b) as the result of existing equations contradicts such conclusion. 44

59 Chapter Three: Model Development In the previous chapter, we introduced our modified augmentation of the linear constitutive equations to be applicable to the materials nonlinear regimes. As mentioned, a domain switching model is required to complete the constitutive model. In addition, we developed a numerical solution procedure for the constitutive model to be used to predict the nonlinear response of polycrystalline piezoceramics. In this chapter, we develop a complete model to be used for the prediction of the nonlinear response of polycrystalline piezoceramics. First, we discuss the process of domain switching in single crystals and then introduce proposed models for the switching rate. Second, we develop a Finite Element model for polycrystalline piezoceramics. Finally, we elucidate the numerical procedure employed to obtain response solutions for a given loading input. 3.1 Switching model Experimental data from single crystalline piezoceramics have not been employed to design the switching model. We decide to first focus on single crystals as this gives us the advantage of investigating the mechanism of switching without the complication of grain orientations and inter-grain interactions exhibited by polycrystalline ceramics. Each 45

60 grain in a polycrystalline material can be considered as a single crystal. To design the switching model, experiments conducted on single crystals can be used. In this section, we first review the main features of domain switching in single crystalline materials. Subsequently, we introduce three new switching models developed based on these features. Each model can predict some of these features Experimental data in single crystals We review experimental observations reported in literature for a single crystal of barium titanate. The observations can be employed in developing switching models even when we are modeling other piezoceramics possessing the same perovskite structure. The reason is that the switching mechanism is similar in materials with the same crystal structure. In addition, not all piezoceramics have a single crystal form. The reason is that the grain size of some piezoceramics, such as PZT, is too small and it is not possible to manufacture a single crystal big enough to conduct tests on. Several researchers have conducted experiments to study single crystals of barium titanate [ ]. Merz et. al. [ ] are among the earliest researchers who studied single crystals of barium titanate. They investigated the effect of crystal size and load strength on the switching process. They also studied the generated current when an electric field is applied and held constant over time. In such cases, the current generated can indicate the rate of switching. Shieh and his group [ ] focused on the domain switching in a single crystal under a combination of mechanical and electrical loads. Referring to the aforementioned experimental works, we can summarize the following main features of domain switching in tetragonal piezoelectric single crystals possessing the perovskite structure: 46

61 In tetragonal ceramics, six types of domain can coexist in a single crystal, as shown in Figure 3.1. Adjacent domains are separated by domain walls. The domain walls are categorized by the angle of spontaneous polarization of the domains on each side, 90 and 180 degrees. Switching can take place in two different scenarios: nucleation of new domains or growth of a domain through domain wall movement, as shown in Figure 3.2. If a single crystal is initially poled in the opposite direction to the applied field, domain switching will start with a few cells at some critical points aligning to the electric field to originate new domains. At the critical points local forces are higher due to crystal defects or boundary effects. After nucleation, these new domains grow at their boundaries. The rate of switching depends on the area of walls and the speed of wall motion. When new domains are formed, the area of the walls is very small, Figure 3.2.b. Consequently, the rate of switching is small. As domains grow, their wall area increases until new domains coalesce, Figure 3.2.c. As a result, the rate of switching increases. When new domains that have aligned with the applied load coalesce and start joining together, the area of the walls reduces and so does the rate of switching, as shown in Figure 3.2.d. When all domains finally switch, the rate of switching reduces to zero, Figure 3.2.e. This is known as saturation. Internal loads can trigger switching even if external loads are below coercive strength. This effect is known as intra-granular effect. The rate of switching is higher with a larger applied load for all load strengths. 47

62 Material properties depend on domain polarization direction. Hence, domain switching affects material properties, such as elastic, piezoelectric and dielectric parameters. Figure 3.1: Six types of domains in tetragonal single crystals. The arrows show the direction of spontaneous polarization in each domain. Adjacent domains are separated by domain walls. The type of a domain wall is defined by the angle of spontaneous polarization of domains on each side, 90 and 180 degrees. 48

63 (a) (b) (c) (d) (e) Figure 3.2: Switching process in single crystals. a) A crystal is initially poled in the opposite direction of the applied field. b) Nucleation: A few cells at some critical points start to align to the electric field and originate new domains. c) New domains grow at their boundaries, and consequently the area of walls and the rate of switching increases. d) New domains coalesce so the area of walls and the rate of switching start decreasing. e) All the cells have switched leading to saturation Experimental data of creep response In Section 3.1.1, the main features of domain switching are summarized by focusing on single crystals. These features are very beneficial in understanding domain switching dynamics and consequently in design switching rules. However, they cannot provide the required information leading to quantitative switching. They are qualitative 49

64 and not related to the single crystals of the desired material. In the pursuit of model accuracy with quantitative description, we focus on the results of creep experiments to capture key characteristics related to time-dependent and rate-dependent responses. Zhou [62, ] conducted a thorough experimental investigation of creep effects in PIC151, the piezoelectric material we used in our simulation. He reported creep results of piezoceramics under both electrical fields and mechanical stresses. Although the reported results are related to the macroscopic states of polycrystalline materials, they can be beneficial in designing and tuning our switching models. Referring to his works, we can summarize the following features: Contributions of domain switching to the creep response depend on the magnitude of the applied load and the volume fraction of domains ready to switch. Creep in piezoceramics is of a primary type. This means that the creep rate continually decreases over time. The response follows an exponential curve. The values of creep response at different values of the applied load indicate the rate of switching at different loads Switching model A switching model predicts the evolution of the domains in a single crystal in order to determine the spontaneous strain and polarization under specific mechanical and electrical loading conditions. Based on the observations summarized in the previous section, we propose for the switching model to describe the rate of domain switching. Therefore, our constitutive models can be categorized as rate-dependent models predicting the time-dependent response of piezoceramics compounds. 50

65 Inspired by Huber s model [94], our switching models are based on the following assumptions. First, mechanical stress ij and electrical field E i are considered to be uniform in the crystal, i.e. the same in all domains. Second, the crystal is tetragonal and therefore it has six types of domains. v I represents the volumetric fraction of the domain type I to the volume of the crystal, with I 1,2,..., 6 for tetragonal crystal structures. Third, spontaneous strain and polarization as well as other material properties are described by the volumetric averages over the crystal. Given the total number of domain types, M 6, the number of possible switching among these M domains, is N M ( M 1) 30. Here the process is considered to be unconstrained and therefore any domain can switch to any other domain. In particular, switching from a domain type I to another domain type J, denoted as switching type reduces the volume fraction of domain type I, I v, and increases v J. Let α ω represent the rate of switching of type. Therefore, the rate of volume fraction of domain type I, I v, can be expressed as follows: N I Iα α ν A ω, I 1,2,..., M. (3.1) α1 where I A is the M N connectivity matrix. A I indicates that the activation of switching type increases the volume fraction v I I of domain type I if A 1, decreases it if A I 1, and does not affect this volume fraction if A I 0. The matrix I A for tetragonal crystal is given by Huber et al. [94]. 51

66 Consequently, the resultant spontaneous strain and polarization are given by the volumetric average over the whole crystal. If n I i is the unit vector in the polarization direction of domain type I, then ε ε ν nn, s I I I ij 0 i j I P Pν n. s I I i 0 i I (3.2) where ε 0 and P 0 represent the magnitude of spontaneous strain and polarization component in the polarization direction, respectively. We employ the dyadic product of the unit vector n I i to define the spontaneous strain tensor in Eq. ( 3.3). In addition, a E ε material property L { C, e, χ }, i.e. elastic, piezoelectric, and dielectric tensors, can ijkl ijk ij be calculated based on its values in different domains as follows: I I L ν L. (3.3) I Equations ( 3.2) and ( 3.3) follow the fact that material properties can be expressed in the volume fractions v I. It is also notable from Eq. ( 3.1) that the kinetics of a volume fraction can be defined by the rates of switching between different domain types, f. Therefore, the performance of switching models is related to how we define the rates of switching, f. It becomes apparent that the main features of domain switching in a single crystal depend on how we define the rates of switching. According to the features summarized in 3.1.1, we can assume that the rate of switching is related to the applied load as well as the domain wall area. The latter is directly related to the volume fraction. We consider the 52

67 switching rate as the product of two separate functions: G 1 and G 2 representing the effects of the applied load and the domain wall area, respectively. We call G 1 the loading function and G 2 the saturation function: α I ω G ( E, σ ) G ( v ) (3.4) 1 i ij 2 Let and ij be small changes in polarization and strain due to type Pi switching, the dissipative energy is: G E i P i ij ij (3.5) We also define critical driving force G c as Gc EcP 0 (3.6) where E c is the coercive electrical field and P 0 is the magnitude of spontaneous polarization. G c defined in Eq. ( 3.6) is actually the electric work by coercive electric field per unit volume for a 90 switching. Notice that, in a 90 switching, the change of electric displacement is P. 0 In this thesis, 180 switching is considered as two consecutive 90 switchings. In the following three sub-sections, we describe our proposed relations for G 1 and G2 to match the experimental evidence listed in the previous section The first model This model is a modified version of Hubers s model. Huber et al. [94] introduced a method to calculate α ω for incremental volume fraction. Kamlah et al. introduced a modified version of Huber s model by considering intra-granular effects [80]. We 53

68 propose the following model in order to consider the effects of domain saturation. Our model is based on calculating decremental volume fraction. Inspired by Kamlah s model [80], We consider the following relation for G 1: n1 α α G G G1 Gc Gc 1 G G α α G c G c (3.7) where n is a creep exponent. In this relation, the switching rate for loads smaller than the coercive strength depends on the value of the creep exponent, but for the higher loads the rate does not change. Consequently, we need to involve creep experiments for loads smaller than the coercive strength to choose an appropriate value of n. We consider n 40 for our numerical simulations. Figure 3.3 depicts the loading function for n 40 with a dashed line. In order to predict saturation in domains, we can define function G 2 simply as the volume fraction of the shrinking domain type: G 2 I ν (3.8) Therefore, as domain I shrinks, the rate of switching reduces and gradually becomes zero as illustrated in Figure 3.4. This means that the switching from domain I to domain J depends on the volume fraction of the domain shrinking in this process. 54

69 1 0.8 Loading Function, G st Model 2nd and 3rd Models 0 0 1G c Dissipative Energy for Switching Type, G Figure 3.3: Loading function, G 1, vs. dissipative energy associated with switching type α. The loading function of the first model cuts off at G1 1, for loads higher than the coercive strength, while the loading function for the second and third models increase with a slower rate by increasing the applied load and gradually approach 1. 2G c It is important to note that only a positive switching rate is considered. A switch from domain I to domain J has a positive switching rate. This is equivalent to a switching from domain J to domain I with a negative rate. 55

70 2.5 Saturation Function, G st and 2nd Models 3rd Model Volume Fraction of a Domain, I I Figure 3.4: Saturation function, G 2, vs. volume fraction of a domain ν. The saturation functions of the first and second models linearly decrease when domain I shrinks, while the saturation function in the third model increases to I the point where ν 0.79 and then reduces. The response of a single crystal, subjected to an applied electrical field or mechanical stress can be computed as follows. After calculating G 1 and G 2 from Eqs. ( 3.7) and ( 3.8), the rate of switching can be computed from Eq. ( 3.4). Then the rates of change of all domain volume fractions are obtained from Eq. ( 3.1), and the spontaneous strain and polarization in the single crystal are computed through Eq. ( 3.2) The second model The first model can predict the intra-granular and saturation effects. However, it assumes a constant rate of switching for all the fields over the coercive strength. This 56

71 does not agree with the observation mentioned in 3.1.1, which suggest that the rate of switching increases with increasing loads. Consequently, this model cannot predict the response of piezoceramics subjected to loads higher than the coercive strength. The other limitation of this model is that it cannot describe the nucleation effect, mentioned in Section As suggested by the experimental clues, we define a new function for G 1 as: α 1 G G 1 1 tanh c G a b 2 G c (3.9) Figure 3.3 illustrates this loading function for a 10 and b 11, and shows that it takes a value between 0 and 1. This loading function, Eq. ( 3.9), is actually the logistic function. As a result, the value of G 1 does not cut off at 1, and therefore Equation ( 3.9) can predict the dependence of switching rate on the load above the coercive field, as shown in Figure 3.3. The difference between these two loading functions is shown in Figure 3.3. The parameters a and b can be employed for tuning the switching model. In order to compare the effect of the new loading function, we use the same saturation function, Equation ( 3.8) The third model Finally, we attempt to address the nucleation effect in domain switching. The switching process starts with forming small domains, known as nucleation. As the domains grow, the switching rate increases until the new domains start to join together. Afterwards, the rate of switching reduces until all domains are saturated. To consider this 57

72 effect, we, inspired by Kolmogorov and Avrami statistical theory, consider a new saturation function: G 2 m I c c ν ( m 1) exp I I ν ν ( m1) (3.10) Figure 3.4 depicts this saturation function for m 3 and c It peaks at I around ν 0.75, meaning that at this point the new domains start coalescing. According to Figure 3.4, at around a volume fraction of 0.4, it merges with the saturation function I defined earlier, meaning that all models have the same saturation function for v (3.9). In this model we employed the same loading function as the second model, Eq. 3.2 Finite Element model The previous section dealt with the modeling of domain switching in unconstrained single crystals. In polycrystalline ceramics, each grain can be considered as a single crystalline material. However, the macroscopic response of polycrystalline ceramics is influenced by two additional effects. The first effect is due to the orientation distribution of the lattice axes of the grains. The ceramic shows an average behavior. The second effect is inter-granular interaction, including the generation of mechanical stress due to strain incompatibilities, which may constrain or enhance the switching of domains. The Finite Element Method offers a natural means to take into account both effects by considering each element of a Finite Element mesh to be a grain with its own orientation of lattice axes. Grain to grain interaction is then imposed by solving the Finite Element equations [80]. 58

73 In this section, we review the mechanical and electrical governing equations of piezoceramics. Then, a Finite Element model is developed to solve these governing equations based on the applied loading conditions. The Finite Element model considers the new constitutive relations introduced in the second chapter. The solution procedure of the Finite Element formulation is also discussed. The unknowns in the Finite Element problem are the nodal values of displacement and electrical potential Governing equations We briefly review the governing equations of piezoceramics considering the following assumptions for the boundary value problem: mechanically dynamic, electrically quasi-static, and isothermal. Throughout this section, standard index notation is utilized with summation implied over repeated indices and the subscript (), j representing differentiation with respect to the x j co-ordinate direction. Consider a continuum material of volume V bounded by the surface S. The mechanical and electrical balance equations are, according to the equation of balance of linear momentum and Gauss law, b u ij, j i i in V (3.11) and v Di, i q in V (3.12) where, ij denotes the symmetric Cauchy stress tensor, b i is the body force per unit volume, is the mass density of the material, u i is the displacement, D i is the electric 59

74 displacement, and q v is the free charge per unit volume. The superposed double dot represents a second derivative with respect to time. and The infinitesimal strain, ij, and quasi static electric field, E i, are 1 ij u i, j u j, i (3.13) 2 E i, i (3.14) where, is the electrical potential. On the surface S t, traction t i and stress ij are in equilibrium according to: ijn j ti on t S (3.15) Here ni is the outward normal vector of the surface S t. In addition, the free charge per unit area q s residing on the surface S q has the following relation with the electrical displacement, D i : D n i i s q on q S (3.16) Finally, Equations ( 3.11) to ( 3.16) can be written in the following weak form V u u dv i i V V D E dv i i dv ij ij V V v q dv bu dv i i S S t u ds i s q ds i (3.17) formulations. The weak form of the governing equations will be employed in the Finite Element 60

75 3.2.2 Formulations We use the standard scalar potential formulation which results in similarities in the mathematical structure governing the distribution of mechanical and electric fields [108, 118]. For simplicity, we use vector-matrix notation to express our Finite Element model. In this form, the displacement and electrical potential are calculated by interpolating from the associated nodal values. u N u u n N n (3.18) where u n and n represent the nodal values of displacement and electric potential u respectively. Matrices N and N include the shape functions. From Eqs. ( 3.13) and ( 3.14), strain and electric field are the gradients of displacement and electric potential fields: where u B and B are B u u n E B n x u z u B N 0 z y 0 y z 0 x 0 0 y x 0 (3.19) (3.20) 61

76 x B N y z The governing equations and the proposed constitutive equation in the second chapter eventually yield to the following Finite Element equations: where u T s u T u T m u K uu u K u B dv N b dv N ds t V T s T v T s K u u K B D dv N q dv N q ds uu K, K T and u K u V V V S S (3.21) K are refered to as the mechanical, piezoelectric, and electrical stiffness matrices, respectively; and m represents the mass matrix. u T E u u T K uu B C B dv, K u B e B dv V V T T u T K u B e B dv, K B B dv V V (3.22) u T u m N N dv V Equation ( 3.21) denotes the standard formulation of the Finite Element problem for piezoceramics. The first terms on the right hand side represents the contributions of spontaneous stress s and electric displacement D s. These quantities are the results of spontaneous strain and polarization: s E s 1 C e P s D s e T s P s (3.23) If spontaneous strain and polarization remain constant, as in linear piezoelectricity, it is reasonable to assume that their values are equal zero. Considering 62

77 them causes bias values of displacement and electric potential. In this case, Eq. ( 3.21) converts to the standard Finite Element formulation for linear piezoceramics. However, the focus of this research is on the nonlinear behaviors due to the spontaneous strain and polarization. We consider the spontaneous stress and electric displacement in Eq. ( 3.23) Solution procedure To solve Eq. ( 3.21) we re-write it in a generalized matrix-vector format: M X K X R (3.24) where X and R denote the generalized displacement and force vectors. M and and, u X u,r r u r K are generalized mass and stiffness matrices, respectively, r and m 0 M, K 0 0 K uu Ku Ku K r represent mechanical and electrical forces. u T s u T u T r u B dv N b dv N t ds V V T s T v T s r B D dv N q dv N q ds V V S S (3.25) (3.26) (3.27) In this work, we study the quasi-static case where the rate of loading is very slow. Consequently, the acceleration component is very small compared to stiffness term and can be neglected. Therefore, Eq. ( 3.24) reduces to: X R K (3.28) 63

78 The generalized force vector R can be calculated when the boundary conditions and the internal configuration are given. To find the displacements and electric potentials at each nodal point, Eq. ( 3.28) should be solved for X. There are a variety of numerical methods to solve this type of equation. However, a major issue is that the stiffness matrix K is ill-conditioned, which is caused by the fact that mechanical and electrical properties have qualities with very different orders of magnitudes. In particular, for a kind of PZT, the elements of the E elasticity matrix C that result ink uu are several hundred Giga-Pascal (GPa). On the other hand, the dielectric coefficients that generate K are in the range of several nano-farad per meter. To resolve this issue, we re-scale the variables and rewrite the Eq. (3.28) as: K X R (3.29) where u X 6 10, K 6 K uu 10 Ku 6 K u K 10,R r u r 10 6 (3.30) This means that solving Eq. ( 3.29) results in the displacements at each node while the electric potentials are 6 10 times the actual levels. 3.3 Numerical simulation Earlier in this chapter we have introduced the switching model describing the evolution in a single crystal, and the Finite Element model required for polycrystalline materials. These two models are coupled, i.e. the switching model needs the stress and 64

79 electric field at each grain calculated by Finite Element and the Finite Element model requires the spontaneous strain and polarization of each grain. In this section, we discuss how these two models are synchronized in order to simulate a polycrystalline piezoceramic Material properties We choose PIC151 for our numerical simulation because a wide range of experimental data for this material are available in the literature with which we can compare our results. In addition, its high permittivity, high coupling factor and high piezoelectric charge constant make it a standard material for actuators and sensors. PIC151 is a soft PZT provided by Physik Instrumente (PI). The average grain size in this compound is 6μm and its breakdown field is almost 5 MV / PIC151 are provided by PI and listed in Table 3.1. m. The material properties of The values listed in Table 3.1 are the material properties of PIC151 when it is polarized in the positive Z direction. However, polarization is not always aligned in this direction. In this case, the material properties can be calculated by applying a rotation matrix to the polarization vector. In the next section, we explain how this rotation matrix can be determined. Table 3.1: Material properties of PIC151 [PI] Elastic Constants E C11 E C12 E C MPa 78.91MPa 79.83MPa 65

80 E C33 E C44 E C MPa 24.53MPa 27.80MPa Piezoelectric Constants e C/ m e C / m e C/ m Dielectric Constants ε 8 χ F / m ε 8 χ33 Switching Properties F / m P C/ m ε Coercive strength ( E ) 1.00 MV / m c Grain orientations and polarization directions Polycrystalline materials consist of a large number of grains oriented randomly in three-dimensional space. A local coordinate system ( x, y, z) parallel to the lattice axis can represent the orientation of each grain. The material property and spontaneous state tensors are constructed using the local coordinate system. A global coordinate system is required for the response to be measured. We employ Euler s angles to relate the two reference frames. The local coordinate system is specified by three Euler angles (,, ) with respect to a global Cartesian coordinate frame ( X, Y, Z ), as shown in Figure

81 Figure 3.5: Global and local coordinate frames. The local coordinate ( x, y, z) representing the orientation of each grain is specified by three Euler angles (,, ) with respect to a global Cartesian coordinate ( X, Y, Z ). angles: The rotation matrix of each grain can be expressed according to the Euler s R Rot(, z) Rot(, y) Rot(, z) E cos( )*cos( )*cos( ) -cos( )*cos( )*sin( ) cos( )*sin( ) -sin( )*sin( ) - cos( )*sin( ) cos( )*cos( )*sin( ) - cos( )*sin( )*sin( ) sin( )*sin( ) +cos( )*sin( ) cos( )*cos( ) -cos( )*sin( ) sin( )*sin( ) cos( ) ( 3.31) As mentioned, the material properties are usually reported assuming that the material is poled in the positive z direction. However, each grain consists of different domains with different polarization directions. Therefore, another rotation matrix is 67

82 required to transform material properties within each grain. We employ the following matrices for the six possible domains in a tetragonal crystal structure: Positive x: R D , Negative x: R D Positive y: R D , Negative y: R D (3.32) Positive z: R D , Negative z: R D Consequently, the total rotation matrix for each domain can be found as the result of these two rotations. Rot R R (3.33) E D Afterwards, the material property tensor can be transformed employing the total rotation matrix Rot Piezoelectric specimen For numerical calculations, a piece of rectangular cuboid shape with a size of 5mm 5mm 15mm is considered. Then, the specimen is discretized with linear eight node cubes. As shown in Figure 3.6, the edges of the specimen and the cubes are parallel to the global Cartesian coordinate ( X, Y, Z ). Therefore a total of 375 elements, 540 nodes, and 2160 degrees of freedom are modeled in our Finite Element model. Each cubic element serves as a grain in a polycrystalline structure. A local coordinate frame is attached to each element corresponding to the grain crystalline orientation. As noted 68

83 earlier, a rotation matrix is defined for each element to transform the properties and variables to the global frame. Although theoretically employing finer mesh size can improve the accuracy of Finite Element model, increasing number of elements requires more computational effort. In this problem that the specimen geometry and loading conditions are not so complex, and a structured mesh is utilized, the above mentioned number of elements would be sufficient. The Finite Element model has been validated by comparing with a commercial software-package in Chapter 4. In addition, to simulate a polycrystalline material, we need to consider a number of grains with a wide variety of orientation. The above mentioned elements can ensure considering different orientations of grains. Y Z X Figure 3.6: The piezoelectric specimen and corresponding elements. A specimen of 5mm5mm 15mm with cubic elements is considered for Finite Element calculations. 69

84 3.3.4 Procedure Figure 3.7 illustrates the steps we follow to calculate the nonlinear responses of piezoceramics. As depicted in Figure 3.7, the process requires known values for material properties, initial conditions, and boundary conditions. We consider the material properties of PIC151, listed in Table 3.1, and generate a set of three random values of Euler angles for each element. We have utilized the same set of Euler s angles for all simulations. In most cases, we start with unpoled material in random orientation and equal volume fraction for all domains. Mechanical and electrical boundary conditions for each problem should also be defined. The solving procedure starts with rotation of the material properties and calculating the matrices in the Finite Element relation, Eq. ( 3.28). Subsequently, we can solve for u and ; and consequently calculate E and. With these values, the switching model can determine the new volume fractions and spontaneous properties. Then, the strain and electric displacement of each element can be found by averaging over all domains. To find the strain and electric displacement of the polycrystalline, we average over all elements. To continue to the next step, the material properties and the corresponding stiffness and force matrices should be updated and then the same steps are repeated. 70

85 Material properties: Cdκε,,, 0, P 0 Initial conditions: Euler s angles, volume fractions Boundary conditions: Mechanical and Electrical I v Material properties of all domains in each element, Eqs. ( 3.2) and ( 3.3) Stiffness matrices and force vectors for each element, Eqs. ( 3.25) and ( 3.26) Solve for u and of all nodes, Eq. ( 3.28) E and of each element Switching Model Domains volume fractions in each element, Eq. ( 3.1) Spontaneous strain and polarization, Eq. (3.2) Strain and electric displacement for each element Strain and electric displacement of the polycrystalline End of modeling Next Step is reached No Stop Figure 3.7: Flowchart of the applied nonlinear constitutive model. Finite Element model and switching model should together form the nonlinear constitutive model. 71

86 Chapter Four: Model Validation and Results In this chapter we employ our developed model to predict the response of piezoceramics subjected to a wide range of mechanical and electric loading conditions. The material we consider is the PIC151stack introduced in Chapter 3. We evaluate the performance of our model for three different cases. First, we verify our model for the linear behavior of piezoceramics. Then after tuning the parameters in the switching models, we move on to the nonlinear response in single crystal materials. We compare our results against the qualitative features of domain switching reviewed in Chapter 3 to verify the proposed switching models. Finally, we consider polycrystalline piezoceramics subjected to different mechanical and electrical loading conditions. In addition, we employ the experimental data from the literature to illustrate that our model can quantitatively match some experimental data. In particular, the creep results under electrical loading are employed for parameter tuning in the switching models. 4.1 Linear response As mentioned in Chapter 1, applications of piezoceramics are often limited to a small loading range to avoid nonlinearities. Although nonlinear behavior can still be seen in this regime, only a small portion of domains would undergo switching and consequently the linear response is dominant. We can utilize the developed model to investigate the linear response by neglecting the nonlinear portion of the response and employ the Finite Element model without a switching model. In this case, our model 72

87 reduces to the common Finite Element model employed for linear piezoelectric materials. A major purpose of this approach is to numerically verify our Finite Element code written in the MATLAB software. As a first test, we employ the Finite Element model without the switching model. Consequently, the material properties and spontaneous states do not change over time, and the response is not time-dependent for a quasi-static situation. Thus, spontaneous components in the Finite Element relations, Eq. (3.23), can be neglected resulting in linear behaviors. We conduct this test to validate our Finite Element model and to illustrate that our model can be reduced to a linear model in special cases. We consider a specimen of PIC151 material with dimensions of 5mm 5mm 15mm as introduced in Chapter 3. This specimen is poled and driven in the longitudinal direction by an electric field of 2 MV/ m. Then we employ our Finite Element model to solve for the nodal values of electric potential and displacement in three dimensions. Figure 4.1 shows the total displacement ranging from zero for points on the bottom surface to mm for the corner points on the top surface. In Figure 4.1, the arrows depict displacement direction of each point. An obvious check is performed doing calculations by hand. Based on the linear constitutive relation and considering zero stress, strain in the longitudinal direction is 33 ( 33 / E E ε e C33) E3, where e 33 and C 33 are the elastic and piezoelectric constants in this direction. The values of these constants are listed in Table 3.1. Assuming uniform strain, deformation at the top surface is given by the strain times the length of the specimen. For an electric field of 2 MV/ m, the deformation will be 0.14mm, which agrees our Finite 73

88 Element result. Note that in the Finite Element model we consider boundary effects and also lateral deformations. x Figure 4.1: Total displacement under a constant co-axial electric load. An electric field of 2 MV/ m is applied to a 5mm5mm 15mm specimen of poled PIC151 in the longitudinal direction. Maximum displacement is mm which happens at the top corner points. To validate our results, we also solved the same problem with the commercially available software-package, COMSOL Multiphysics. The model in COMSOL contains 2041 tetrahedral elements. Figure 4.2 illustrates the total displacement ranging from zero at the bottom surface to mm for the corner points at the top surface, i.e mm difference. For further comparison, the difference between these two results is illustrated in Figure 4.3. Figure 4.3 shows that our results agree with the results obtained with 74

89 COMSOL with the difference of less than mm and proves validation of our Finite Element model. In addition, this proves that the number of elements we consider in the Finite Element model is sufficient. Figure 4.2: Total displacement of the piezoelectric stack subjected to electric field by COMSOL Multiphysics. An electric field of 2 MV/ m is applied to a 5mm 5mm 15mm specimen of poled PIC151 in the longitudinal direction. The maximum displacement, mm, happens at the top corner points. 75

90 Figure 4.3: The difference between the results from Finite Element and COMSOL Multiphysics. The difference between the total displacement calculated by Finite Element model, Figure 4.1, and by COMSOL Multiphysics, Figure 4.2. The difference is less than mm for the whole domain. The results shown in Figure 4.1 and 4.2 are generated with the assumption that the piezoelectric material is uniform and completely poled in the longitudinal direction. The symmetry of the vector fields in Figure 4.1 illustrates such uniformity. However, in a polycrystalline material, grains have different crystal orientations. Consequently, in a poled material, the spontaneous polarization of all grains is not parallel. Our Finite Element model can consider this diversity of orientations. The orientation of each element is described by a set of Euler angles, as explained in Chapter 3. Figure 4.4 shows 76