ANALYSIS OF PIEZOELECTRIC CYLINDRICAL A C T U A T O R S AND 1-3 PIEZOCOMPOSITE UNIT C E L L S By Yue Chen B. Eng., Shanghai Jiao Tong University, P. R. China M. Sc. in Engineering Mechanics, Shanghai Jiao Tong University, P. R. China A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 2003 © Yue Chen, 2003 Library Authorization In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Name of Author (please print) Date (dd/mm/yyyy) Title of Thesis: Degree: M(U4^r « | AfptW Department of Me.eJuucr-Cini The University liversity of British Columbia Vancouver, BC Canada ScJ&iUJL Y e a r : V*Jrnuui47K^ ™ « XgQlf Abstract Adaptive structures incorporated with sensors and actuators are increasingly used in many engineering applications. Actuators used in adaptive structures are made out of piezoelectric ceramics, shape memory alloys (SMA), electrorheological fluids (ER fluids) or magnetostrictive materials. Among the many types of piezoelectric actuator elements, the cylindrical shape is widely used in practical applications involving fuel injectors, atomic force microscopes, high-precision telescopes, etc. In addition, the piezoelectric phase of piezocomposites is made out of cylindrical rods or fibers. Piezoelectric materials are very brittle and stress/electric field concentration at electrodes and other discontinuities often contribute to mechanical or dielectric breakdown. The study of electromechanical field of a cylindrical piezoelectric element and a composite unit cell with a piezoceramic core surrounded by a polymeric shell is therefore important to the understanding of failure of cylindrical actuators and design of piezocomposites for maximum electromechanical coupling. This thesis presents a comprehensive theoretical study of homogeneous piezoelectric cylinders and a unit cell of 1-3 piezocomposites. The governing equations for coupled axisymmetric electroelastic field in a transversely isotropic piezoelectric medium are established in terms of the displacements and electric potential. The general solutions of the governing equations are obtained in terms of a series of Bessel functions of the first and second kind. Several boundary-value problems are solved, and a computer code is developed to compute the electroelastic field in solid and annular cylinders for different aspect ratios, electromechanical loading and material properties. The salient features of the electroelastic field are identified. The effective properties of a 1-3 piezocomposite are studied under hydrostatic loading for different fiber volume fractions and polymer and ceramic properties. Optimum fiber volume fractions for maximum electromechanical coupling are determined for different ceramic-polymer combinations. ii Table of Contents Abstract ii List of Tables v List of Figures vi List of Symbols . xi Acknowledgements 1 2 3 xiii Introduction 1 1.1 General 1 1.2 Introduction to Piezoelectric Materials 3 1.3 Piezoelectric Constitutive Relations 5 1.4 Application of Piezoelectric Materials 9 1.5 Literature Review 11 1.6 Scope of the Current Work 14 Analytical Solution of Finite Piezoelectric Cylinders 15 2.1 Problem Description 15 2.2 Field Equations 15 2.3 General Solution of Governing Equations 17 2.4 General Solution of Electroelastic Field 22 2.5 Solution of Cylinders Subjected to Electromechanical Loading 24 2.6 Solution of Solid Cylinders 35 Analytical Solution of a 1-3 Piezocomposite Unit Cell 38 3.1 Problem Description 38 3.2 Governing Equations and General Solution of the Polymer Matrix 42 3.3 Piezocomposite Cylinder Subjected to Hydrostatic Loading 46 iii 4 5 Numerical Results and Discussion 54 4.1 Comparison with Elastic Cylinders 54 4.2 Electroelastic Field of a Solid Cylinder under Mechanical Loading 63 4.3 Electroelastic Field of a Solid Cylinder under Electrical Loading 71 4.4 Electroelastic Field of an Annular Cylinder 81 4.5 Response of a Unit Cell of a 1-3 Piezocomposite 89 Conclusions 99 5.1 Summary 99 5.2 Suggestions for Future Work 101 Bibliography 103 Appendix A 107 iv List of Tables Table 1.1 Linear piezoelectric constitutive relations Table 1.2 Material constants of piezoelectric ceramics with hexagonal symmetry .. .9 Table 4.1 Material properties of selected piezoceramics 63 Table 4.2 Material properties of polymer phase of a 1-3 piezocomposite 90 v 6 List of Figures Figure 1.1 Concept of an adaptive structure 2 Figure 1.2 Remanent polarization due to poling 4 Figure 1.3 (a) Direct and (b) converse piezoelectric effects 5 Figure 1.4 Variation of (a) electric polarization with electric field, (b) strain with electric field and (c) stress with strain 8 Figure 1.5 Perovskite unit cell 8 Figure 1.6 (a) Tube actuator and (b) multilayer stack actuator 10 Figure 1.7 Schematic of a 1-3 piezocomposite 11 Figure 2.1 Annular piezoelectric cylinder and the coordinate system 15 Figure 2.2 Annular piezoelectric cylinder under vertical loading 24 Figure 2.3 A piezoelectric finite cylinder under applied electric charge loading 33 Figure 3.1 Piezoelectric rods of a 1-3 piezocomposite 39 Figure 3.2 1-3 piezocomposite materials and typical unit cells 40 Figure 3.3 Circular cylinder approximation of a unit cell 41 Figure 3.4 Relationship between rod volume fraction and unit cell dimensions 42 Figure 3.5 Geometry of a unit cell of a 1-3 piezocomposite and coordinate system 43 Figure 4.1 Comparison of series expansion of q(r)=\-2r Figure 4.2 Series expansion of the discontinuously applied loading at r=0 for different M values Figure 4.3 55 56 Series expansion of the discontinuously applied loading at r=0.005b for different M values 57 Figure 4.4 Series expansion of the applied loading at different r values 57 Figure 4.5 Nondimensional vertical stress at selected points at z*=l of a solid cylinder for different values of N Figure 4.6 58 Nondimensional vertical stress profiles along z-axis (r=0.005b) of a solid cylinder (h/b=l) for different values of M and N 59 Figure 4.7 Comparison of solutions for stresses of a solid magnesium cylinder 60 Figure 4.8 Comparison of stresses of a hollow magnesium cylinder 62 Figure 4.9 Piezoelectric cylinder subjected to uniform vertical pressure 63 vi Figure 4.10 Nondimensional vertical stress profiles along the r-axis (z=0) 64 i Figure 4.11 Nondimensional hoop stress profiles along the r-axis (z=0) Figure 4.12 Nondimensional vertical electric field profiles along the r-axis (z=0) .... 65 Figure 4.13 Nondimensional vertical stress profiles along the z-axis (r=0) Figure 4.14 Nondimensional vertical displacement profiles along the z-axis (r=0)... 66 Figure 4.15 Nondimensional vertical electric displacement profiles along the z-axis (r=0) 64 65 66 Figure 4.16 Nondimensional vertical electric field profiles along the z-axis (r=0) .... 67 Figure 4.17 Nondimensional vertical stress profiles along the r-axis (z=0) of PZT-5H cylinders for different h/b ratios Figure 4.18 Nondimensional hoop stress profiles along the r-axis (z=0) of PZT-5H cylinders for different h/b ratios Figure 4.19 72 Nondimensional vertical stress profiles along the r-axis (z=0) under applied electric charge loading Figure 4.25 73 Nondimensional vertical electric field profiles along the r-axis (z=0) under applied electric charge loading Figure 4.27 73 Nondimensional hoop stress profiles along the r-axis (z=0) under applied electric charge loading Figure 4.26 71 Piezoelectric finite cylinder subjected to electric charge at the top and bottom ends Figure 4.24 70 Nondimensional vertical electric field profiles along the z-axis (r=0) of PZT-5H cylinders for different h/b ratios Figure 4.23 70 Nondimensional vertical electric displacement profiles along the z-axis (r=0) of PZT-5H cylinders for different h/b ratios Figure 4.22 69 Nondimensional vertical stress profiles along the z-axis (r=0) of PZT-5H cylinders for different h/b ratios Figure 4.21 68 Nondimensional vertical electric field profiles along the r-axis (z=0) of PZT-5H cylinders for different h/b ratios Figure 4.20 68 74 Nondimensional vertical stress profiles along the z-axis (r=0) under applied electric charge loading vii 75 Figure 4.28 Nondimensional vertical displacement profiles along the z-axis (r=0) under applied electric charge loading Figure 4.29 75 Nondimensional vertical electric displacement profiles along the z-axis (r=0) under applied electric charge loading Figure 4.30 76 Nondimensional vertical electric field profiles along the z-axis (r=0) under applied electric charge loading Figure 4.31 76 Nondimensional vertical stress profiles along the r-axis (z=0) of PZT-5H cylinders under applied charge loading for different h/b ratios Figure 4.32 77 Nondimensional hoop stress profiles along the r-axis (z=0) of PZT-5H cylinders under applied charge loading for different h/b ratios Figure 4.33 78 Nondimensional vertical electric displacement profiles along the r-axis (z=0) of PZT-5H cylinders under applied charge loading for different h/b ratios Figure 4.34 78 Nondimensional vertical stress profiles along the z-axis (r=0) of PZT-5H cylinders under applied charge loading for different h/b ratios Figure 4.35 79 Nondimensional vertical electric displacement profiles along the z-axis (r=0) of PZT-5H cylinders under applied charge loading for different h/b ratios Figure 4.36 ....79 Nondimensional vertical electric field profiles along the z-axis (r=0) of PZT-5H cylinders under applied charge loading for different h/b ratios 80 Figure 4.37 Annular cylinder under vertical pressure Figure 4.38 Nondimensional vertical stress profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading Figure 4.39 82 Nondimensional vertical electric field profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading Figure 4.41 82 Nondimensional vertical electric displacement profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading Figure 4.40 81 83 Nondimensional vertical stress profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading viii 84 Figure 4.42 Nondimensional vertical electric displacement profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading Figure 4.43 84 Nondimensional vertical electric field profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading 85 Figure 4.44 Piezoelectric annular cylinder subjected to electric charge loading 85 Figure 4.45 Nondimensional vertical stress profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under electric charge loading Figure 4.46 Nondimensional vertical electric displacement profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under electric charge loading Figure 4.47 91 Nondimensional vertical stress profiles at the centre of a unit cell (a/b=l/6, h/b=5/3) Figure 4.54 91 Nondimensional shear stress profiles along the interface of a unit cell (a/b=l/6, h/b=5/3) Figure 4.55 92 Effective hydrostatic piezoelectric constant dh as a function of volume fraction of piezoelectric phase Figure 4.56 94 Effective hydrostatic voltage constant gh as a function of volume fraction of piezoelectric phase Figure 4.57 90 Nondimensional radial displacement profiles at the centre of a unit cell (a/b=l/6, h/b=5/3) Figure 4.53 89 Nondimensional vertical displacement profiles of the top surface (z=h) of a unit cell (a/b=l/6, h/b=5/3) Figure 4.52 88 Nondimensional vertical electric field profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under electric charge loading Figure 4.51 88 Nondimensional vertical electric displacement profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under electric charge loading Figure 4.50 87 Nondimensional vertical stress profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under electric charge loading Figure 4.49 86 Nondimensional vertical electric field profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under electric charge loading Figure 4.48 86 94 Variation of dh with piezoelectric rod aspect ratio (Vp25%) ix 95 Figure 4.58 Variation of dh with volume fraction of piezoelectric phase for different matrix materials (a/2h=0.1) Figure 4.59 Variation of g„ with volume fraction of piezoelectric phase for different matrix materials (a/2h=0.1) Figure 4.60 97 Nondimensional vertical electric field profiles along the z-axis of a unit cell (a/b=l/2) Figure 4.62 96 Variation of dhgh with volume fraction of piezoelectric phase for different matrix materials (a/2h=0.1) Figure 4.61 96 98 Nondimensional vertical displacement profiles along the z-axis of a unit cell (a/b=l/2) 98 List of Symbols a Inner radius of a cylinder or radius of the piezoelectric phase of a 1-3 piezocomposite b Outer radius of a cylinder or radius of the polymer matrix of a 1-3 piezocomposite b Radius of the applied load on the top and bottom surfaces of the cylinder Cy Elastic constant £> Electric displacement component in the /-direction D Nondimensional magnitude of electric charge applied on the top and bottom 0 0 surfaces of the cylinder d Hydrostatic piezoelectric coefficient of a piezocomposite d Piezoelectric strain constant d Effective hydrostatic piezoelectric coefficient of a piezocomposite Ej Electric field component in the /-direction e Piezoelectric stress constant h i} h y g Hydrostatic voltage coefficient of a piezocomposite g Piezoelectric voltage constant g Effective hydrostatic voltage coefficient of a piezocomposite h Half height of a cylinder I Modified Bessel function of the first kind and order n h v h n J n Bessel function of the first kind and order n K Modified Bessel function of the second kind and order n k Electromechanical coupling coefficient n q tJ q 0 Piezoelectric constant Hydrostatic pressure applied on the two ends and the lateral surface of a 1-3 piezocomposite xi q(r) Nondimensional normal pressure applied on the top and bottom surfaces of the annular cylinder w, Displacement component in the /-direction Y Bessel function of the second kind and order n n Strain component By Dielectric constant <p Electric potential <p Nondimensional electric potential q> Potential function of the matrix phase of a 1-3 piezocomposite qj Nondimensional potential function of the matrix phase of a 1-3 piezocomposite A\ Roots of the characteristic equation of a piezoelectric cylinder or the piezoelectric phase of a 1-3 piezocomposite v,. Roots of the characteristic equation of the matrix phase of a 1-3 piezocomposite <jy Stress component y/ Potential function of a piezoelectric cylinder or the piezoelectric phase of a 1-3 piezocomposite ys Nondimensional potential function of a piezoelectric cylinder or the piezoelectric phase of a 1-3 piezocomposite xii Acknowledgements I extend my sincere appreciation to my advisor, Dr. R. K. N. D. Rajapakse for his guidance, continuous support, valuable time and criticisms throughout the course of my graduate study at the University of British Columbia. I am indebted to Dr. Rajapakse for providing me with financial support from his research grants. I would like to thank the members of my thesis committee, Dr. M. Gadala and Dr. O. Kesler, for reading my thesis and making valuable comments. I also want to thank my parents and my friends. Their understanding, encouragement and support enabled me to complete this work. xiii Chapter 1 INTRODUCTION 1.1 General The concept of an adaptive structure [1,2,3] has recently been introduced as an effective approach for shape, vibration and acoustic control. An adaptive structure differs from a traditional passive structure in that active elements are incorporated into the system. Active elements consisting of actuators and sensors can be attached to a structure after assembly or they can be embedded in structural elements during fabrication. As the mass and stiffness of a structure are distributed along its spatial dimensions, the active components could also be distributed to allow for a large number of sensors and actuators for structural control. The failure of a single actuator or sensor becomes much less critical to the overall performance of an adaptive structure with a distributed network of sensors and actuators. In addition, such a structure also possesses an integrated closed-loop control system for active control of the actuators based on the interpretation of signals from the sensors (Fig. 1.1). Adaptive structures are used in a variety of engineering applications that require shape, vibration and acoustic control of mechanical systems to meet certain performance goals. Many different types of sensors and actuators have been developed in recent years for incorporation in adaptive structures. For strain sensing, piezoelectric sensors, fiber optic sensors or strain gauges have proven to be useful. Actuators made out of piezoelectric materials, shape memory alloys (SMA), electrorheological fluids (ER fluids) and magnetostrictive materials are often proposed for adaptive structure applications [4]. At present, sensor technology is well developed but actuators based on piezoelectric materials or SMA are relatively new and require further research to 1 understand their behaviour under complex electromechanical or thermomechanical loading. One useful material for actuation of mechanical structures is the nickel-titanium SMA. Shape memory alloy actuators provide large forces for their relatively small size and weight during a transformation between two material microstructural phases. The phase transformation of SMA can be accomplished by heating or cooling the material across a transition temperature. Below the transformation temperature of SMA, the material is relatively soft and deformable and can be mechanically deformed by several percent without damage. This deformation can be reversed by heating the material above the transformation temperature, where the material has a higher strength and cannot easily be deformed. However, the range of transformation temperatures and lack of control over the cooling time limit the application of SMA actuators in some practical situations. Fig. 1.1 Concept of an adaptive structure. Piezoelectric materials are also commonly used for sensing and actuation of adaptive structures [5,6]. These materials generate a charge in response to a mechanical deformation. Conversely, they produce a mechanical strain under an applied electric field. When used in adaptive structures, actuation is controlled by an applied electric field. Piezoelectric materials are available in a wide variety of shapes and sizes and can be distributed along a structure without greatly increasing its mass. Piezoelectric 2 materials are very brittle and this together with their electric fatigue performance is a major concern in applications. Piezoelectric coupling coefficients are very small for most materials and the strain (stroke) produced by an electric field may not exceed a few microns. 1.2 Introduction to Piezoelectric Materials The literal translation of the term "piezo" comes from the Greek word "piezin", which means "to press". More specifically, the term refers to the ability to create electricity by pressure, and conversely, the development of mechanical strains by electricity. The discovery of piezoelectricity is credited to Curie brothers. In 1880, the two brothers demonstrated that when a stress in the form of a weight was applied to quartz and other crystals, positive and negative charges were developed on certain portions of the surface. The charge was proportional to the pressure exerted and disappeared when the pressure was withdrawn. They later confirmed that a change in the dimension of a crystal occurred when a voltage was applied. Twenty natural crystal classes in which the piezoelectric effect occurs and all eighteen possible macroscopic piezoelectric coefficients were later defined. Most of the classical piezoelectric applications with which we are now familiar with (microphones, accelerometers, ultrasonic transducers, flexural actuators, signal filters, etc.) were conceived and applied to practice following World War I. The discovery of poled ceramics with relatively large piezoelectric coefficients expanded the applications of piezoelectric materials to a wide variety of applications, such as powerful sonar devices, fuel injectors, ultra sensitive microphones, etc. When a material is brought into an external electric field E, each particle of the material is subjected to an "internal field" E n proportional to the applied field E. In some materials, under the influence of an internal field the particles are moved over a long distance. Such materials are called conductors and an equilibrium position is not 3 reached in them before thefieldstrength becomes zero at all points inside the material. In the case of insulators (dielectrics) [7], only very small displacements of the charged particles occur when an electric field is applied. The forces acting on the charges bring about a small displacement of the electrons relative to the nuclei, as the applied field tends to shift the positive and negative charges in opposite directions. The reason that this displacement is limited is that the electrons are bound to the nuclei and reactive forces that are proportional to the displacements exist. Then it is said that the dielectric is polarized by the applied field: it is brought into a state of electric polarization. Now consider a simple system consisting of only two point charges + e and —e at a distance a [Fig. 1.2 (a)]. Such a system is called an electric dipole. Its moment is called the electric dipole moment. There are two kinds of dipole moments, a permanent dipole moment and a temporary dipole moment. In the first form of polarization, as mentioned above, under the influence of an electric field, the positive and negative charges are moved apart, and a temporary dipole moment is produced. Another kind of polarization is due to the fact that an applied field tends to permanently direct the dipoles. Atfirst,the directions of domains which are regions of local alignment formed by adjoining dipoles in a material are random [Fig. 1.2 (a)]. Under an applied electric field, domains that are most nearly aligned with the applied field expand in the direction of thefield[Fig. 1.2 (b)]. When the electric field is removed, most of the dipoles are locked into a configuration of near alignment and the material (a) Directions of domains before (b) During polarization polarization Fig. 1.2 Remanent polarization due to poling. 4 (c) Remanent polarization after electric field moved possesses a non-zero remanent polarization [Fig. 1.2 (c)]. After being polarized, the material shows the piezoelectric effect. There are two kinds of piezoelectric effects [8]. The first is called "direct piezoelectric effect" and it means that electric polarization is produced by mechanical stress. In the case of "converse piezoelectric effect", a crystal becomes strained when an electric field is applied. That means a voltage applied across a piezoelectric crystal causes it to expand or contract. If the direction of the applied field is the same as that of the polarization, the material expands. The material contracts if the directions are opposite to each other. Fig. 1.3 shows the two types of the piezoelectric effects. (b) Fig. 1.3 (a) Direct and (b) converse piezoelectric effects. 1.3 Piezoelectric Constitutive Relations The ability of a piezoelectric material to convert mechanical energy into electrical energy, and vice versa, is expressed by the electromechanical coupling coefficient k [9]. Consider a single piezoelectric element subjected to external pressure. Under this condition, a portion of the applied mechanical energy is converted to an electric charge that appears on opposite faces of the element. The charged piezoelectric element is 5 analogous to a charged capacitor. The balance of the applied mechanical energy is converted into strain energy, much as in the case of a compressed spring. When the pressure is removed, the element returns to its original shape and the electric charge disappears. Based on this description, k can be defined as ^2 _ mechanical energy converted to electrical energy applied mechanical energy The same basic relationship holds true when an electric charge is applied and ^2 _ electrical energy converted to mechanical energy applied electrical energy Under a low applied field, piezoelectric behaviour can be described by simple linear relationships involving stress ( o ) , strain (€), electric displacement (Z>), and - electric field (E) [8]. From different thermodynamic functions, four types of constitutive relations can be derived. Table 1.1 shows the different types of linear piezoelectric constitutive relations. Table 1.1 Linear piezoelectric constitutive relations. Independent variable Piezoelectric relation \a = c e-q D [E = -q€ + p D q -form \e = s a + d E D = da + e"E d -form D e,D T e E T \ e = s a+g D D a,D T \<r = c e-e E I D = ee + s E E e,E Identification T £ 6 g -form e-form In the above relations, c and s denote the elastic stiffness matrix and elastic compliance matrix respectively, d, piezoelectric strain constant matrix, measures the strain in a free piezoelectric material for a given applied field; e, piezoelectric stress constant matrix, measures the stress developed by a given field when the piezoelectric materials is clamped; g, piezoelectric voltage constant matrix, measures the open-circuit voltage for a given stress; and q measures the open-circuit voltage for a given.strain, fi and e represent impermittivity and permittivity of the piezoelectric material respectively [10]. The superscript E means under constant electric field; D means under constant electric displacement; o means under constant stress; E means under constant strain; and the superscript T denotes transpose of a matrix. The number of non-zero independent material constants in each relation is determined by the symmetry class of a material. Note that each type of relation can be obtained from a simple linear transformation of one of the remaining relations. The linear constitutive equations are developed on the assumption that the electric displacement is directly proportional to the electric field. This assumption is, however, true only for a homogeneous and isotropic dielectric material under a low electric field [11]. Nonlinearities of piezoelectric materials are clearly demonstrated in the experimental investigation performed by Crawley and Lazarus [12]. At large driving voltages, the piezoelectric coefficients are seen to increase, resulting in a nonlinear relationship between the electric field and the induced strain. Typical non-linear relationships between electric polarization and electric field, strain and electric field and stress and strain are shown in Fig. 1.4. Crystals can be divided into thirty-two classes on the basis of their symmetry properties. Of these thirty-two classes, twenty are piezoelectric and essentially anisotropic. Since the piezoelectric effect exhibited by natural materials such as quartz, tourmaline, Rochelle salt, etc. is very small, polycrystalline ceramic materials such as 7 E (a) (b) (c) Fig. 1.4 Variation of (a) electric polarization with electric field, (b) strain with electric field and (c) stress with strain. BaTiOy and Lead Zirconate Titanate (PZT) have been developed with improved properties. The most important chemical structure of a piezoelectric ceramic crystallite is the perovskite, (AB0 ). 3 This structure can be described by a simple cubic unit cell with a large cation on the corners, a smaller cation in the body center, and oxygen atoms in the centers of the faces as shown in Fig. 1.5. AT. 7 1 ! !/ ! A o-4- • A: t 1 >--I— 6 1 2+ •B: Zr , Ti* A+ °C: —o *• Pb + O2 7 Fig. 1.5 Perovskite unit cell. Most piezoelectric ceramics posses hexagonal symmetry. That means in these materials there are three equivalent axes lying 120° apart in a plane perpendicular to the axis of material symmetry. One of these three equivalent axes is chosen as x, the axis of material symmetry is taken as z and y is taken in a right-handed axis system. These piezoelectric materials are transversely isotropic with the axis of material symmetry 8 parallel to the poling direction. They have only ten independent nonzero material constants as shown in Table 1.2. Table 1.2 Material constants of piezoelectric ceramics with hexagonal symmetry. Elastic stiffness (c C| 2 Cn Piezoelectric constants Cl3 0 0 0 .3 0 0 0 0 0 0 0 0 0 44 0 C C 33 0 0 0 C44 0 0 0 0 0 0 0 0 C 0 0 0 e 0 0 g 0 0 0 V 3) 3. 0 0 is 0 0 0 0 0 e Dielectric constants 0 e 33y Piezoelectric ceramics such as PZT share the material characteristics of many common ceramics. As such they are very strong in compression, but rather weak in tension. In addition, they are quite brittle materials. 1.4 Application of Piezoelectric Materials Piezoelectric materials have been used in a wide variety of applications, including as transducers in underwater sonar devices, and as tweeters in audio speakers. PZT ceramics have been used for structural excitation of turbomachinery and other components. They have also been incorporated into truss members and electrically tuned to increase the overall passive damping of structures. Piezoelectric polymer films, a low modulus polymer material, have been used as actuators and sensors in both open and closed loop applications. A piezoelectric actuator converts an electrical signal into a precisely controlled physical displacement which can be used to finely adjust precision machining tools, lenses, or mirrors. Piezoelectric actuators are receiving considerable attention in the recent years, with the advent of new applications such as scanning-probe microscopy, 9 adaptive telescopic mirrors, active suspension systems for automobiles, etc. They are available in a wide range of sizes and shapes such as thin and thick discs, solid and annular cylinders, etc. Modern instruments such as scanning tunneling and atomic force microscopes universally use piezoelectric tube actuators to generate the fine motions required [13]. Multilayer stack actuators are often used in disk drives, printer heads and fuel injectors. Fig. 1.6 shows the schematic of a piezoelectric tube actuator and a multilayer stack actuator. A sensor converts a physical parameter, such as acceleration or pressure, into an electrical signal. In some sensors the physical parameter acts directly on the piezoelectric element and in other devices an acoustic signal establishes vibrations in the element which in turn is converted into an electrical signal. Often, in an adaptive system, a visual, External electrode Ceramic layer Internal electrode (a) (b) Fig. 1.6 (a) Tube actuator and (b) multilayer stack actuator. audible, or a physical response is produced to a signal from a sensor (e.g., an automobile seatbelt locks in response to a rapid deceleration). The purpose of manufacturing composites is to produce a new material with overall properties superior in some fashion to the constituent materials. Addition of a second material increases the number of parameters available to customize a composite material. Therefore, not only can the designer specify the materials, but also their relative amounts, shapes, orientations, and manufacturing methods. In this way, it is possible to 10 make use of the best attributes of each material to get structural properties not achievable by a single material. For example, structural composites have made use of strong stiff fibers and softer epoxy matrix materials. This combination guarantees strength and stiffness through the fibers while the matrix provides load transfer capability, shear properties, and conformability. The combination of piezoelectric ceramics with another material has been coined "piezocomposites" [14,15,16]. It can be used to maximize electromechanical coupling. The fibers (rods) of piezocomposites are made of piezoelectric materials and the surrounding matrix is usually a polymeric material, such as a high-modulus epoxy or an elastomer of a relatively low modulus. The maximum coupling in piezocomposites is achieved by transferring the applied stresses to the composite's piezoelectric component in a manner that most nearly approximates the piezoceramic's maximally coupled stress patterns. A typical piezocomposite is named on the basis of its connectivity. For example, a 1-3 piezocomposite refers to a one-dimensionally connected PZT phase, and a threedimensionally connected polymer phase. This type of piezocomposites exhibits higher sensitivity and lower mechanical losses than monolithic piezoelectric ceramics in the case of acoustical applications. Fig. 1.7 shows the schematic of a 1-3 piezocomposite. Polymer matrix Piezoelectric rod F i g . 1.7 Schematic o f a 1-3 piezocomposite. 1.5 Literature Review Among the many types of piezoelectric actuator elements, the cylindrical (solid and hollow) shape is used in a broad spectrum of practical applications such as fuel 11 injectors, atomic force microscopes, high-precision telescopes, etc. In addition, the piezoelectric phase in piezocomposites is made out of cylindrical rods or fibers. Therefore, the study of electroelastic field in a piezoelectric cylinder under combined electromechanical loading is one of the fundamental problems in adaptive structures technology. Electric field and stress concentrations occurring in certain regions of a cylindrical piezoelectric element subjected to electromechanical loads could lead to dielectric breakdown, electrode delamination and fracture. Furthermore, tensile stresses due to loading could lead to tensile cracking due to the low tensile strength of piezoceramics. Stress analysis of elastic cylinders under various boundary conditions is one of the fundamental problems in elasticity and has a rich history [17-20]. Levine and Klosner [17] analyzed an infinite cylindrical shell subjected to band loads. Atsumi and Itou [18] considered an infinite circular cylinder with a spherical cavity. Kasano et al. [19,20] considered the stress distribution in a cylinder due to both ring loads and two diametrical point loads. All the above analyses are focused on infinite circular cylinders, such as infinite solid cylinders or hollow cylinders. The three-dimensional problem of a finite cylinder, however, is much more complicated than that of an infinite cylinder [21-31]. A finite isotropic cylinder under axial compression was analyzed by Pickett [21] using a multiple Fourier-Bessel series solution. A similar analysis for a constrained elastometric cylinder under end compression was done by Moghe and Neff [22]. Power and Childs [23] presented a solution for a finite isotropic circular bar subjected to axi-symmetric tractions and/or displacements on either or both ends. The elastic analysis of transversely isotropic finite cylinders with stress-free lateral surfaces was considered by Vendhan and Archer [24] by using a displacement potential. Okumura [25,26] used the generalized Elliott's solution to analyze a transversely isotropic, short hollow cylinder subjected to an outer band load and examined the effects of anisotropy on stress distribution. More recently, the analytic solutions for a finite transversely isotropic solid cylinder under different surface loads 12 were presented by Wei and Chau [27,28,29]. They used the Lekhnitskii's stress functions [30,31] to uncouple the equations of equilibrium. Moreover, a new Fourier-Bessel series expansion of the stress function was proposed so that all boundary conditions can be exactly satisfied. It was found that the stress distribution was non-uniform along the axis of loading, and a local maximum tensile stress was developed near the point loads. Wei and Chau [29] showed that the material anisotropy changed the magnitude of stress, but basically did not influence the shape of stress distribution along the axis of loading. Montgomery and Richard [32] presented an analytical model to predict the mechanical response of a composite cylinder that is composed of a solid right circular cylinder surrounded by a cylindrical tube of a different material. Their model can be readily extended to analyze the unit cell of a 1-3 piezocomposite. The past studies on elastic cylinders lay a strong foundation for the advancement of theoretical analysis of piezoelectric cylinders. Wang and Zheng [33] obtained the general solution of the three-dimensional governing equations for a transversely isotropic piezoelectric medium. Their solution can facilitate the derivation of analytical solutions for some practically useful problems involving piezoelectric materials. Rajapakse and Zhou [34] used Fourier integral transforms to derive a theoretical solution for a long piezoceramic cylinder subjected to axially symmetric electromechanical loading. Their solution examined the effects of coupling between mechanical and electric fields, magnitude of actuating stresses and zones of potential mechanical and dielectric failure. In addition, their solution serves as a basic building block for the study of load-transfer mechanisms, delamination and sensory regions of adaptive composite elements containing piezoceramic fibers. Li and Sottos [35] developed an approximate solution for a piezocomposite finite cylinder to predict the load transfer and the effective hydrostatic voltage coefficient of 1-3 piezocomposites. They investigated the influence of matrix stiffness, interlayer stiffness, rod aspect ratio, and rod volume fraction. 13 1.6 Scope of the Current Work Based on the above introduction and literature survey, it is found that piezoelectric cylinders are one of the most common types of actuators. A rigorous study of their electromechanical behaviour has not appeared in the literature. Furthermore, the basic cylinder model serves as a building block for the analysis of more advanced problems such as piezocomposites. Therefore, the main objectives of this thesis are to conduct a comprehensive theoretical analysis of piezoelectric finite cylinders, examine the salient features of the electroelastic field in a cylinder and develop an accurate theoretical model for a unit cell of a 1-3 piezocomposite. Chapter 2 presents the details of derivation of the theoretical solution for homogeneous piezoelectric solid and annular cylinders. The basic governing equations of a piezoelectric cylinder are presented in the cylindrical coordinate system. A set of potential functions are used to transform the governing equations expressed in terms of displacements and electric potential to a set of Laplace equations. The general solutions for Laplace equations are obtained in terms of the Bessel functions of the first and second kind. The applied loading of a cylinder is expressed in terms of a Fourier-Bessel series to determine the solutions for arbitrary functions appearing in the general solutions. Chapter 3 is devoted to the analysis of hydrostatic response of 1-3 piezocomposites. A unit cell of a piezocomposite is represented by a composite cylinder, which is composed of a solid piezoelectric inner cylinder bonded to an outer elastic annular cylinder. Perfect bonding and electrically impermeable boundary conditions are assumed along the interface. Chapter 4 presents the numerical results for various cylinder geometry, material and loading conditions. The influence of material properties, loading and cylinder dimensions on the distribution of elastic and electric fields in a cylinder is investigated. Effective properties of a 1-3 piezocomposite are investigated for different volume fractions of the piezoelectric phase, piezoelectric and composite material combinations, and piezoelectric rod aspect ratios. Conclusions along with the recommendations for future work are presented in Chapter 5. 14 Chapter 2 ANALYTICAL SOLUTION OF FINITE PIEZOELECTRIC CYLINDERS 2.1 Problem Description In this chapter, the derivation of analytical solutions for solid and annular piezoelectric cylinders is presented. Fig. 2.1 shows a typical cylinder of inner radius a, outer radius b and height 2h. A cylindrical polar coordinate system (r,0,z) as shown in Fig. 2.1 is used with the z-axis along the axis of symmetry of the cylinder. The cylinder is made out of a piezoelectric material with hexagonal symmetry or a poled ceramic with the poling direction parallel to the z-axis. Loading is assumed to be axi-symmetric and applied at the top and bottom surfaces and/or the inner and outer cylindrical surfaces. All field variables such as displacements, electric potential, etc. are independent of the circumferential coordinate 6 due to the axi-symmetric geometry and loading of the problem. x Fig. 2.1 Annular piezoelectric cylinder and the coordinate system. 2.2 Field Equations The constitutive equations for piezoelectric materials which are transversely isotropic or poled along the z-axis can be expressed as [36], 15 <7 = cuerr rr +c e +c e -e E l2 ge n zz 3l ^=c e„+c e^+c e -e £ 1 2 n 1 3 z z (2.1) z 3 1 (2.2) z <r= = c e +c e„ + c e - e E l3 rr 33 l3 zz i3 (2.3) z a = 2cjE. -e..E rz 44 rz 15 (2.4) r v A =e ,e +e 6^ + 6 3 3 6 , , + £ £ =2c, €„ 3 5 where a (j rr 31 3 3 ' (2.5) z , e , £>,. and E denote the components of stress tensor, strain tensor, electric t displacement vector and electric field vector respectively; c,,, c , c , c 12 elastic constants under zero or constant electric field; e , e 31 constants; and e , and e x 33 x3 and e 15 33 and c M are are piezoelectric are dielectric constants under zero or constant strain. 33 The field equations of a piezoelectric material undergoing axi-symmetric deformations about the z-axis can be expressed as —!r. —s. _z + dr ^ 3r ^- = 0; —-+——+—^- = 0 + r dz ^ + + 9z ^ r = dr (2.6) r dz (2.7) 0 The strain-displacement relations are _ du or r . _ M r • _ r du . dz _1 2 2 V 3z (2.8) oV where u and « denote the displacements in the r - and z - directions respectively. r z The relationship between the electric field E (i = r,z) and the electric potential 0 j can be expressed as E =- -±; dr d r E =-^2 (2.9) dz Substitution of eqns (2.1) - (2.5), (2.8) and (2.9) in eqns (2.6) and (2.7) yields the following governing equations expressed in terms of the elastic displacements u and u r and the electric potential <j>. 16 z (d\ {dr ldu r dr 1 ^ r J r d\ dz , ,d u drdz 2 z (2.10) (e +e )|^ = 0 31 15 fd\ {dr \du] r dr J ^l }_dl) dr r dr d\ dz + + du "15 dr z ( d <p + e 33 r oV J 3 3 + 2 19^ 2 2 d\ , dz (c 2 dr 2 \ du\ r dr J 2 (2.11) e 2 f r J_]l 33^T =0 "dz r "15 {d\ ldu ) {drdz r dz J ( jd\ 1 du {drdz r dz e | 31 r 15 (2.12) dV -=0 dz 7 2.3 General Solution of Governing Equations In this section, the theoretical solution of the governing equations (2.10) - (2.12) is derived. To facilitate the derivation of the analytical solution, the following potential function representation is introduced. u- , r u -/c, , z dr (2.13) <p- dz e,, dz where y/{r,z) denotes a potential function, and £, and k are unknown constants to be 2 determined. Substitution of eqn (2.13) in eqns (2.10) - (2.12) leads to the following set of governing equations expressed in terms of y/{r,z). d y/ Ady/ + k + K (c + c )+k (e , + c ) ] | ^ = 0 dr r dr r 2 4 2 (c A:, + c + c + e k ) 44 13 M Xi 2 13 M 13^ dr 2 r dr 17 2 3 15 (2.14) (2.15) (e rC, + e 15 3 1 +e e, ,& ) l5 2 oV (2.16) r oV 2 Before proceeding to solve the above governing equations, it is prudent to nondimensionalize all field variables. Typical values of elastic, piezoelectric and dielectric constants of piezoelectric materials are different by many orders of magnitude and this could lead to precision and numerical instability problems during the calculation of numerical solutions [37]. For example, the material constants of PZT-4 have the following orders: 10 N/m ; 10 Cy ~ e ~ 10° C/m ; 2 e ~ 1 (T F/m 2 9 g tJ The coordinates r and z and the displacements u and u are nondimensionalized r z by the outer radius b which is set as the nondimensional unit length parameter. The stresses and elastic constants are nondimensionalized by c . The electric displacements M and piezoelectric constants are nondimensionalized by e . For convenience, the 3] nondimensional coordinates, displacements, stresses, electric displacements, elastic constants and piezoelectric constants are denoted by the same symbols without loss of generality. In addition, the following new nondimensional quantities are introduced. (h = (2.17) (h • w = — • £ =^-£ • £ =^-£ 631 h A 2 ' 2 3 2 Using the nondimensional quantities, eqns (2.13) and (2.14) - (2.16) are changed to: dy/ u.=^r-; ~dr~ d y7 r 2 dr 2 (k + c x 1 3 | u =k —; z 2 dz 1 dy/ r dr az \[l+* (l+c )+* (l+ I 1 3 + l+e k {^-+}+[c k ^ dr or r dr or ) l5 (2.18) </> = k t 2 2 33 { 18 2 e i 5 (2.19) )]^ =0 +e k 33 2 dz =0 (2.20) r^2 dyr dy/ + [e &, £ k ] dr 1 33 dr 2 i3 2 2 —0 (2.21) It can be seen that all nondimensional material properties appearing in the above governing equations have similar orders of magnitude and d y7 \dy7 dr dr r constants. As the terms 2 and + 2 and k are dimensionless 2 d w^are not identically equal to zero, a r 2 dz 2 nontrivial solution of eqns (2.19) - (2.21) exists if and only if, 1 + ft + C 1 3 ) *1 + I 1 + !5 ) 2 e _ k C 33^1 + g 33^2 +1 + C | + e k c 3 n i5 l + (l + c, )^ +{l + e )k _ 3 l5 2 e 33^1 +\+ =X (2.22) -=x (2.23) 2 ^33^2 e -£ k ]5 u where X is a dimensionless constant to be determined. The eqns (2.22) and (2.23) have three unknowns jt k and X. Eliminating 2 and k in eqns (2.22) and (2.23), the following cubic algebraic equation of X is obtained. 2 n x +n x +n x+n =o 3 x where the coefficients 2 2 3 (2.24) 4 Q. (/ = 1,2,3,4) are constants expressed in terms of material properties and are defined in the Appendix [eqn (Al)]. The three roots of eqn (2.24) are denoted by X (i = 1,2,3) with X^ assumed to be i a positive real number and X and X are either positive real numbers or a pair of 2 3 complex conjugates with positive real parts. For each root of eqn (2.24), the eqns (2.22) and (2.23) yield the solutions for k and k • It is convenient to denote the corresponding x 2 solutions by k (i = l,2;j = 1,2,3) where the subscript j identifies the corresponding root lj X. t 19 In view of the three roots A, obtained from eqn (2.24), the eqns (2.19) - (2.21) yield the solutions for three potential functions y/. (i = 1,2,3) governed by, ^ 3 A or r or , az + (i = U 3 ) 0 (2-25) Eqns (2.18) can be rewritten in terms of y/. as dr t = k W az + k az J j L + k az az az (2.27) J j L az Combining eqns (2.1) - (2.5), (2.8), (2.9), (2.26) and (2.27), the solutions for stresses and electric displacements can be obtained in terms of yj. [A(2) - A(6)]. Eqn (2.25) can be expressed in the following form ^or| ri or M az^ + + = 0 (/ = 1,2,3) (2.28) i where z. = z/^A\ . ( The solution of eqn (2.28) is given by [38], W, = [AJ (tr)+BY (tr)] [C, cosh{tz )+F sinh(t )] 0 0 i i Zi (i = 1,2,3) (2.29) where J {tr) and Y (tr) are Bessel functions of the first and second kind of zero order, 0 0 respectively [39]; coshitz^ and sinhfe,) are hyperbolic cosine and sine functions, respectively; and t, A, B, C,. and F (/ = 1,2,3) are arbitrary functions to be determined. On the other hand, eqn (2.25) can also be expressed in the following form. a ^ or + 1 a ^ r. dr. + a ^ = 0 ( . = U dz 20 3 ) ( 2 3 0 ) where =^r- rj The solution of eqn (2.30) is given by [38], W = [GJ {sr )+L K {sr )][Pcos{sz)+Rsm{sz)] (/ = 1,2,3) i 0 i i 0 (2.31) i where / (s>;) and K^s^) are the modified Bessel functions of the first and second kind 0 of zero order, respectively [39]; and , G , L , P and R s t = 1,2,3) are arbitrary t functions to be determined. Therefore, the general solution of eqn (2.25) can be expressed as, ^=[A/ (fr)-r-5y>)][^ r / \ / vi r / \ / vi [ G , / (sr,)+(sr )] [Pcos{sz)+R sin{sz)] 0 0 U = 1,2,3) (2.32) t In the present study, only axi-symmetric loading of a cylinder is considered. Therefore the potential function should only contain even functions of z. In addition to the solution given by eqn (2.32), it is necessary to include the potential functions for the radially symmetric plane problem of an annular cylinder and that of a long bar for the completeness of the general solution. Therefore, the complete general solution of the potential function y7 in eqn (2.18) can be expressed as, i=l Z 1=1 X K^olf r)+B Y (t r)]cosh{t z,)+ m IM 0 m m (m=l £ [GJ (S^+L^K,for,)]cos(s z) 0 n n=l (2.33) YlTt where and A , B , G , L , A im im in in 0i (/ = 1,2,3) and B oy are arbitrary functions and t is a constant. These unknown quantities have to be determined from the boundary m conditions. 21 2.4 General Solution of Electroelastic Field The general solution for the complete electroelastic field of a cylinder can be obtained by substituting eqn (2.33) into eqns (2.26), (2.27) and (A2) - (A5). It is convenient to express the general solutions for displacements, stresses, etc. as sum of three parts: (a) the first part denoted by superscript '0' corresponds to the first three terms of eqn (2.33); (b) the second part denoted by superscript ' 1' corresponds to the series containing the Bessel functions of the first and the second kind in eqn (2.33); (c) the third part denoted by superscript '2' corresponds to the series containing modified Bessel functions of the first and the second kind in eqn (2.33). The corresponding solutions are: (2.34) /=i °?M u 0 (2.35) +c -2 )A* = {c -c )\B ,+fl{c n 1=1 u l2 Xl /=i r (2.36) 4 = - 4 I > A ; <ri=0 0 ) 0) 1=1 (2.37) /•=i (2.38) f > [A J {t r)+B Y {t r)}cosh{t ) 4\r,z) = m im x m im x m mZi i'=l m=l (2.39) /=1 m=\ • 3 t^rUiJo <p (r,z) = ±k (1) 2i 1=1 {t r)+Bjr {t r)\,wh{t z,) m 0 m m (2.40) m=l • ("c,, + X;)tj«{t r)+ m (c,, -c )-/,(t r) r 12 1=1 m=l m A i m B,_ 22 + \cosh{t z,) m (2.41) Z v, Z'* U J 0 (tmr)+ im 1=1 (2.42) B Y {t r)]co h{t ) im 0 m S mZi m=l ~Z Z£ k ^ . M i=l (2.43) + BJT {t r)]sinh{t ) X m mZi m=l -Eft Z'i k.- . ( ' - ' H ) (2.44) 7 D%,z) = i=l m=l M]«wA('.*,) (2.45) (s r, )]cos{s z) (2.46) u \r,z) = -2X I>„ [ G / fc.'iI)+L K [s K )]sin{s z) (2.47) Di\r,z) = 2>, tA (v)+ ; i=l m=l u {r,z)= Z Z VV. fo'i (v.hLJC, {2) 1=1 n n n=l (2 h i'=l 0 (r,z) = {2) 0 in 0 n n n=l (2.48) 3 -Z*2/ Z* fc-Ak>;)+ n r /=! n=l (ci. + Xi U^s„K 0 2 4 {s r,)- (c -c,, ) - * , for,) r H 12 *2>M ="Z ' Z ^ "to» ofa-l)+A**o u cos(s„z) ( - 9) 7 (2.50) i=i «=i (2.51) o®{r.z)i=l D?(r.z): -Zsi>»1^ . (v;)~ A ^ i (v, 7 1=1 D?\r,z)-- n=l n=l - & ! > » [GJo (v,)+« (v, Wv) i=l (2.52) (2.53) n=l The above general solution for electroelastic field in a finite piezoelectric cylinder is a new contribution that has not previously been reported in literature. 23 2.5 Solution of Cylinders Subjected to Electromechanical Loading In this section a hollow cylinder subjected to electromechanical loading is analyzed by using the general solutions derived in the preceding section. First consider the case of a cylinder as shown in Fig. 2.2 under normal stresses applied to the top and Fig. 2.2 Annular piezoelectric cylinder under vertical loading. bottom ends. Assume that all surfaces of the cylinder are electrically impermeable and the inner and outer cylindrical surfaces are stress free. The boundary conditions can be expressed as follows: <7 (a,z) = 0, rr <7 (l,z)=0, rr a„{a,z) = Q, <7„(U)=0, CT {r,±h) = -q{r), (T {r,±h) = 0, zz zr D (a,z) = 0 for r D {U)=0 r for -h<z<h -h<z<h D {r,±h) = 0 for 2 a<r<\ (2.54) (2.55) (2.56) where -q{r) denotes the nondimensional normal pressure applied on the top and bottom surfaces of the cylinder. The boundary conditions expressed by eqns (2.54) - (2.56) have to be used to determine the arbitrary functions A , I M B I M ,G I N and L I N (/ = 1,2,3;/W,M = 1,2,...,<») appearing in the general solutions. Given the complexity of the analytical general solutions, it is important to apply the boundary conditions in a systematic way to solve for the arbitrary functions. 24 First consider the boundary condition G = 0 at z = ±h. Noting that and zr of) vanish at z = ±h, the boundary condition, a =of) +&^} zr to cj }=0- =0> c a n ° reduced e Then using eqn (2.43) and noting that the boundary condition has to be zf satisfied for a < r < 1, the following relationships can be obtained. i>,£4-^('.A)=o; i^fX^M>>>0 (2.57) where h = hj JA\ . { Using eqn (2.57), express A Xm and B 3m in terms of A and A , and B lm 3m Xm in terms of B.2m as A =0( A +a A ; m IM m 2m B =a B +a B 3m lm lm 2m 2m (2.58) 3m where _ ti sinh(t h ) . ^\Sinh{t h ) 2 m _ j) sinh{t h ) & inh{t h ) 2 m 3 2m x lS m m 3 ( 2 5 9 ) y Substitution of eqn (2.58) into eqn (2.43) leads to, Noting that the shear stress boundary conditions on the inner and outer cylindrical surfaces (r = a,l) have to be satisfied for -h<z<h, the following conditions can be established. oSM=o; 4 M=0; ) 2) O £)(i,z)=o; <xiM=0 2) (2.61) Substituting the first condition of eqn (2.61) in eqn (2.60) yields, ^=-44^; j^ _I&A Zm T (, \ Zm im (2.62) B = T /' \ 25 Then substituting eqn (2.62) into eqn (2.60) and imposing the third condition of eqn (2.61) yields the following transcendental equation to determine t . m Next the arbitrary functions G Xn and G can be expressed in terms of G and ln 3n (i = 1,2,3) by considering the second and fourth conditions of eqn (2.61) in eqn (2.51). L in Therefore, G = (2.64) - f Xn (2.65) G =2n J /=i where *i,(r) = W r,{r)=*M (V,): ~ rAaMO-rtiMa)'~ #0 ^ ( i ) (2.66) (i = U,3) (v;) - ^ W K } C Until now, only shear stress boundary condition of the cylinder has been used (a = 0 at r-a,\ and z = ±h) and the unknowns A , A , A , B , G zr Xm 2m 3m Xm Xn and G 2n are expressed in terms of the remaining arbitrary constants appearing in the general solutions. Next, consider the normal stress and normal electric displacement boundary conditions given by eqn (2.56). Substituting the eqns (2.58) and (2.62) in eqn (2.42) yields, fl> W (1)/ \ ^ 2 C O S h ^ ^ )°lm + V2 COSh{t Z m 2 )Jfi +1 2m ( 2 v fiR where ^ M ^ - T H ^ M + ^ M 0 = 0.1) 26 (2.69) Substitution of eqns (2.64) and (2.65) in eqn (2.50) yields, 4 M=-I* 2) YA ) a 2 (v) >cos (2.70) YA°) Next, following an identical procedure, can be expressed as, f [T, cosh{t z )a + r cosh{t z )]B + m x Xm M r . c o s / z^-xi^2 ^z,)^, m 2 m 2 2m - - - -z )]B ^cosh{t + ^ m 3 im and ^ ftfrE, ^ ( a ) / (. r )-T ^ / (. r ) r3^/ (^r ) 0 n 1 2 0 0 n 2 + 0 3 YAa) cos (2.72) \L,_ YAo) Substitution of eqns (2.58) and (2.62) in eqn (2.44) yields the following expression for £)W. (2.73) Then the substitution of eqns (2.64) and (2.65) in eqn (2.52) yields, G + 3a Yi\a) >siw (2.74) Y (o)Si+^(a) 2 Yi(a) where flF (r) = / ftVi(Vi): Ato = £,4*.(v,) 0 = U,3) Following an identical procedure, <jM can be expressed as, 27 (2.75) [a cosh{t z )+ cosh{t z )] -c t H {t r)+ lm 5>. m x m 2 u m 0 (c,, - c m L 1 2 )-#, (t r) m r .+'»\X\<*x cosh{t z )+x cosh{t z )]H [t r) m [a 2m m x 2 m cosh(t z )+cosh{t z )] m x m 3 2 0 m c t H (t r)+ u m 0 m B, + L \X\(hm cosh{t z )+Zi cosh(t z )]H (t r) m x m 3 0 m (2.76) Then using eqns (2.64), (2.65) and (2.49), <^ M=Z r G 2) 0n where j- j^r L 3n+ in in can be expressed as, (2.77) cos(s z) n (/ = 0,1,2,3) are defined in the Appendix [eqns (A7) and (A8)]. All components of stresses and electric displacements involving the remaining boundary conditions are now expressed in terms of the arbitrary functions B 2m and L in , B , G 3m 3n (i = 1,2,3). Assuming that the series involving m and n indices converge for M and N terms, the remaining boundary conditions of the cylinder have to be used to determine the (2M + 4N) unknown arbitrary functions appearing in the general solution. Until now only the shear stress boundary condition is used and the remaining boundary conditions involving radial and vertical normal stresses and electric displacements can be used to determine the (2M+4N) arbitrary functions. Now consider the boundary condition - 0 at' r = a which can be expressed as, . (2.78) Noting that H, (t a) = 0, eqn (2.76) is reduced to m 28 - c,, (a cosh{t z )+ cosh{t z ))+" B + cosh{t z )+z cosh{t z ) lm m x m 2 2m m x 2 m 2 - c,, (a cosh(t z )+ cosh(t z ))+ B 2m m x m 3 ^,or cosh{t z )+z cosh{t z ) 2m m x 3 m •#o('»fl) (2.79) 3m 3 In order to apply the boundary condition given by eqn (2.78) at a constant r value, it is necessary to express the variation of radial stress in the z -direction in terms of identical functions of z . To achieve this, the hyperbolic cosine terms in eqn (2.79) are expressed in terms of a Fourier series of the following form: cosh(t z,) = y® + |>1 cos{s z) (2.80) (i = 1,2,3) 0 m n n=l where (,)_sinh{t h,) (/) m 2t sinh{t h )cos(nn) = m m (2.81) i Substitution of eqn (2.80) in (2.79) makes the z-coordinate dependence of eqns (2.79) and (2.77) identical thus allowing the grouping of terms with similar cosine functions. Then applying the boundary condition given by eqn (2.78) yields the following set of linear relationship between the arbitrary functions. 3 1 £2(c,, + c - 2 X i ) A 7=f X2 0 i + {c -c X2 xx )—B a 0X + (2.82) ,0. •H (t a) = 0 0 ,(3) ^"-H m=\ ( ( 1 ) (3)) [-Cu{^2n,yn+yn)+ZlO: y-+Z ynK 2 (1) '(3) I D '3m i 1=1 29 m f"0Vm J + <3 (2.83) Next, it is straightforward to apply the boundary condition Noting that H, (t ) = 0, <T = rr 0 at r = 1. at r = 1 can be obtained simply by replacing 'a' in eqn m (2.79) by T . Then using eqns (2.80), (2.81) and (2.77), the following linear relationships between the arbitrary coefficients are obtained. 3 £2(c, + c -2 .)A ] n X 0 i + (c -c )B 12 u 0i + (2.84) s l[- ,,k^i +^ )^,«2^I +^ k m , ) 1 ) ) 3 ) c m J (2.85) i=i The boundary condition D =0 at r = a,\ can be expressed as, r (D; +/J>; +D; L=O; O 2 ) ( D ^ + D ^ + D ; ^ ) ^ =O O ) Noting that Z)(' (a,z)=0, Z)J (l,z)=0 and ) l) £>; (a,z)=0; £>J 0) =0, eqn (2.86) reduces to, Di (l,z)=0 2) (2.86) (2.87) 2) Substitution of eqn (2.87) in eqn (2.74) yields the following set of linear relations between the arbitrary functions G + 3n Y\\P) i=i 7\{a) (2.88) tU {a)-im {a)-p,{a)I,„ = 0 x 2 30 y^te-y' «r,(i)-flft(i)^,(i)' w (2.89) L =0 s Next, consider the boundary condition = —q{r) at z = ±h which can be expressed as, (c?V+*~+o^\__ =- {r) ±h (2.90) q In order to apply eqn (2.90), it is necessary to express each term of eqn (2.90) in terms of identical functional variations of the radial coordinate. To achieve this, first expand q(r) into a Fourier-Bessel series of the following form. (2.91) m=l where l{q{r)rdr Ja 0o = f q{r)rH {t r) dr 0 • \-a 2 r\ m '» \ m (2.92) (rH (t r)dr f 2 0 m Ja Then expand the Bessel and the modified Bessel functions of the second kind, i.e. / (s r ) and K (s^) (/ = 1,2,3), in eqn (2.70) in terms of a Fourier-Bessel series as, t 0 /o(v) = ^ + £/i|^ (r.r); ( 0 K (s„ ) = 0 ri S^ m=\ m=l + ts^H (t r) (* = 1,2,3) 0 m where (2.94) Ja ri,) - X . f e J + ^ f e , " ) 2 . [KJMW^ ri„ = Ja 31 (2-95) (2.93) Substitution of eqns (2.36), (2.68), (2.70), (2.91) and (2.93) in eqn (2.90) yields the following linear relationships between the arbitrary coefficients. 4l>A + I M & . ^ +S^L )cosM 2 1=1 2 t m m + [v a cosh(t h, )+v,cosh(t.h,)]B,. }t 2>. 2m m G„ + 7i(«) 2 0 n=l <-{[*W„ cosh(t h )+ u cosh(t h, )\B, m (2.96) =Q in (2.97) 3 cos(n7t) = -Q„ n=l tr\ r.(«) Finally, the remaining boundary condition D = 0 at z = ±h can be expressed as, z ( (0 D + D ( 2 )+ D ( o ) ^ = ( 0 2.98) Substitution of eqns (2.37), (2.71), (2.72) and (2.93) in eqn (2.98) yields the following linear relationships between the arbitrary coefficients. (2.99) 1=1 n=l 'M[W.^*('.*l)+»2<^M'.^)]«!» + [ W . ^ * ( ' » * l ) + « 3 < < « * ( ' - * , ; J ) ] S J . } - (2.100) cos n=l The eqns (2.83), (2.85), (2.88), (2.89), (2.97) and (2.100) constitute a system of linear B, algebraic B, G, L, equations of order (2M + 4N) with arbitrary constants L, and L,„. This system can be solved numerically. In addition eqns (2.82), (2.84), (2.96) and (2.99) can be solved for the remaining four arbitrary 32 constants \ (i = 1,2,3) and B . This completes the solution of all arbitrary coefficients 0i appearing in the general solution for the vertical loading case shown in Fig. 2.2. Electric Loading Condition Now consider the case of a cylinder as shown in Fig. 2.3 which is subjected to electric charge loading applied to the top and bottom surfaces. The boundary conditions can be expressed as follows, <7„{a,z) = 0, <T {a,z) = 0, a {lz) = 0, 0"„(l,z) = O, D {a,z) = 0 rz rr D(r,±h) = \~ ° r D (l,z) = 0 - ~ \ J ,±h)=0, [ 0 b <r<1 D a r b a r r for for <rJr,±h)=0 -h<z<h (2.101) -h<z<h for a<r<\ (2.102) (2.103) 0 where D denotes the nondimensional magnitude of electric charge applied to the top 0 and bottom surfaces of the cylinder. Fig. 2.3 A piezoelectric finite cylinder under applied electric charge loading. The solution of this problem is quite similar to that of a cylinder subjected to vertical pressure on top and bottom surfaces. The only difference is for z = ±h, the vertical stress a zz equal to -D 0 is equal to zero for a $ r < 1 while the vertical electric displacement is for a < r < b . So the eqns (2.96) and (2.97) are changed to, 0 33 45>A t ^t M% + i=l s2 v i=l n (2.104) +Sl%)c s{nx) = 0 O n=l ) + 2 cosh{t h )]B + [v a cosh{t h )+ v cosh(t h )]fl }u m 2 2m x 2m m x 3 COS m 3 3m (2.105) («^) = 0 «=i 1=1 Expand D into a Fourier-Bessel series of the following form, 0 D =D + 0 (2.106) ^D H (t r) 0 m 0 m m=l where ipD.rdr l-a _ f 0 m (2.107) \ m 2 D rH,{t r)dr trHl{t r)dr f m Ja Therefore the eqns (2.99) and (2.100) are changed to, 4l*A,+t^.t M s2 i=i i=l (2.108) G +S(% )cos(nx) = D ] in n 0 n=l cosh{t h )+r cosh{t h )]B + [t a cosh(t h )+r cosh{t h )]g }m x 2 m 2 2m x 2m m x 3 m - i ,(i) ftW4z^fe)_ J /< ^ + ; 2 T I* n=l 2 3 I 3m (2.109) cos{n7t)=-D Z' i=l r 3 n ; ,0) r ftW^lM * '» Tl ZT\ 1 YA ) a Equations (2.83), (2.85), (2.88), (2.89), (2.105) and (2.109) constitute a system of linear algebraic equations of order B ' B , G , L, L 2m 3m 3n u 2n (2M + 4N) with the arbitrary constants and I . These equations can be solved numerically. In 3n addition eqns (2.82), (2.84), (2.104) and (2.108) can be solved for the remaining four 34 arbitrary constants A (i = 1,2,3) and B . This completes the solution of all arbitrary 0l m coefficients appearing in the general solution for the electric charge loading case. 2.6 Solution of Solid Cylinders A solid cylinder is obtained from an annular cylinder when the inner radius is zero. The solution for a solid cylinder can be obtained from the annular cylinder solution by making a few changes. To ensure the regularity of the solution at r = 0, it is necessary to drop the Bessel and modified Bessel functions of the second kind, i.e. Y {t r) and 0 a n ( K { „ i) s r 0 * t n e m logarithmic term of eqn (2.33), which become infinite as the argument r tends to zero. Therefore, the general solution for the potential function corresponding to a solid cylinder is given by, ? = 14* (r - 2zf h 2 1=1 i l E ^ / o ( W ) + I X cosh{t Vl V 1=1 V " = l mZi ]7 (t r)) 0 (2.110) m ) m=l and t is the w-th root of J , (t ) = 0. m m The general solutions for electroelastic field in a solid cylinder are given by eqns (2.34) - (2.53) with i? , B 01 A, 0i A im and G in im and L being equal to zero and only the arbitrary functions in (i = 1,2,3) are left. In the case of a cylinder subjected to vertical pressure, the following six boundary conditions have to be considered, (7 {U) = 0, rr (T„(l,z) = 0, a (r+h) = -q{r), a {r,±h) = 0, z2 zr D {U) = 0 for -h<z<h (2.111) D (r,±h) = 0 for 0<r<l (2.112) r z Using the conditions D (\,z) = 0 and a (l,z)=0, the relations among G , G r rz and G can be obtained as follows, 3n 35 ln 2n Also because a vanishes when z = ±h, zr _ _ d sinh(t h ) 2 m 2 tf sinh{t h ) A 3 " 2 " V m " 2 / ^ ^sinh{t h,) m "3 " \ - m - w } i ) (2.114) j ^sinh{t h ) l m m m t So there are only 3 + N + 2M unknowns, i.e., (/ = 1,2,3), G , A 3 n 2m and A im left. The condition c„.(l,z) = 0 at r = l yields, f (2.115) =0 I=1 V m=l 3 (2.116) =1 m=l 1=1 and P , Q and /? are defined in the Appendix [(A9)-(A11)]. in im /m Now expand the boundary condition <r =-q{r) {z = ±h) into a Bessel-Fourier zz series of the following form, g(r) = Q + 0 (2.117) f Q J (t r) j m 0 m where - Q,=2[q{r)rdr; _ [q{r)rJ*{t r)dr m Q•m = (2.118) f\ m [rJl{t r)dr m Following four equations can be obtained from the boundary conditions a zz = -q{r) and D = 0 at z = ±h. z *l cosh{t h, )A - X m im J ( t \ i \ : \ \cos{nK)G 2 in = ~Qm (2.119) (2.120) i=i 1=1 n=l n i m 36 2>>tcosh(t h,)A -Z m im T n=l oVm 4ZrA + Sir, ^ ^ S ) )\ n i S C =0 \ i ' } ; ; > ( ^ t J 0 A + t (2.121) m) , ( ^ ) G , =0 (2.122) =1 n=\ The eqns (2.115), (2.119) and (2.121) represent (N + 2M) linear simultaneous equations to determine the unknowns G , A 3n (2.120) and (2.122) are solved to determine \ field in a solid cylinder can be determined. 37 2m and A . In addition, eqns (2.116), 3m (z = 1,2,3) - Thereafter the electroelastic Chapter 3 ANALYTICAL SOLUTION OF A 1-3 PIEZOCOMPOSITE UNIT CELL 3.1 Problem Description There are several practical limitations to incorporating piezoelectric materials in adaptive structures. The brittle nature of the ceramics is one of the major limitations. In addition, the limited ability of the piezoceramics to conform to curved surfaces and the higher density of lead-based piezoceramics limit their applications. A composite material consisting of an active piezoceramic fibrous phase embedded in a polymeric matrix phase remedies many of the aforementioned restrictions. The advantages of piezoceramics, i.e. a high structural stiffness and the ability to interact with dynamic systems at frequencies spanning from about 1 Hz well into the megahertz range, can be preserved by using piezocomposite elements in practical applications. In addition to protecting the piezo fibers or rods, the flexible nature of the polymer matrix also allows the material to more easily conform to curved surfaces often found in engineering applications. Piezocomposites offer many distinct advantages with respect to their monolithic counterparts. The multiphase construction yields a more robust actuator that is capable of being added to a lay-up as "active layers" along with conventional fiber-reinforced laminae. While complex arrangements of piezoelectric blocks are required to both bend and twist a structure, piezocomposites yield in-plane actuation anisotropy, allowing a single patch the ability to apply both bending moments and twisting torques. Piezocomposites were originally developed for underwater hydrophone applications in the low-frequency range, but have also been extended to other applications such as ultrasonic transducers for acoustic imaging and biomedical engineering [40-47]. A variety of piezocomposites can be made by combining a piezoelectric ceramic with a passive polymer. These composites are classified according to their connectivity. 38 Type 1 -3 piezocomposites contain piezoelectric rods embedded in a polymer matrix and aligned through the thickness of the device. Fig. 3.1 shows a typical 1-3 piezocomposite. The 1-3 notation specifies the connectivity pattern for this particular arrangement of fibers and the surrounding matrix material. Specifically, the piezoceramic constituent is continuous in one direction, i.e. through the thickness, while the matrix material is connected in all three orthogonal directions. It can be seen that in making a 1-3 piezocomposite, several parameters can be varied: the elastic properties of the polymer phase, the aspect ratio of the piezoelectric rods and the volume fraction/distribution of the piezoelectric rods in a piezocomposite. Fig. 3.1 Piezoelectric rods of a 1-3 piezocomposite. In applications involving hydrophones, a useful measure of the effectiveness of a piezoelectric material is its ability for electromechanical energy conversion, which can be represented by the value of the product of the hydrostatic piezoelectric coefficient, d , h and hydrostatic voltage coefficient, g , of the material [48-51]. Therefore in designing 1h 3 piezocomposites, the primary goal is to maximize the value of d g by varying the h h matrix and fiber properties and fiber volume fraction. A 1-3 piezocomposite consists of a polymer matrix with parallel cylindrical piezoelectric fibers as shown in Fig. 3.2. The fibers are poled in the length direction. The 39 cross section of a typical unit cell of a 1-3 composite can be assumed to be of a square shape [Fig. 3.2 (a)] or of a hexagonal shape [Fig. 3.2 (b)]. Ay y ooooo ooooo ooooo ooooo (a) Cross section of 1 -3 composites and a unit cell of square shape. Ay X z (b) Cross section of 1-3 composites and a unit cell of hexagonal shape. Fig. 3.2 1-3 piezocomposite materials and typical unit cells. In order to simplify the theoretical analysis of a unit cell, a circular cylinder approximation is used here with the square or hexagonal cross-sectional area replaced by a circle of equal area as shown in Fig. 3.3. 40 Fig. 3.3 Circular cylinder approximation of a unit cell. The relations between the dimensions of the square or hexagonal unit cell and that of the circular cylinder model are cf = 3^3 2 and —c 2 h " rcb] and a< — 2 _J2 =7abh A and ^ v3c a< — 2 A (3.1) (3.2) where b and b denote the outer radius of an equivalent circular unit cell corresponding s h to a square and hexagonal unit cell respectively. (a) Square unit cell. 41 1.0 0.8 h 0.6 h 0.4 \- 0.2 \- 0.0 0.0 0.2 0.4 0.6 0.8 a/c„ n (b) Hexagonal unit cell. Fig. 3.4 Relationship between rod volume fraction and unit cell dimensions. The variation of rod volume fraction V in the two models is shown in Fig. 3.4. f 3.2 Governing Equations and General Solution of the Polymer Matrix In this section, the "displacement potential" method which is developed in the previous chapter is extended to analyze a typical unit cell of a 1-3 piezocomposite. The unit cell is made of a single cylindrical piezoceramic rod bonded to a concentric annular polymer cylinder as shown in Fig. 3.5. The radius of the ceramic rod and the outer radius of the elastic matrix are denoted by a and b respectively, and the height of the unit cell is 2h. The polymer matrix is assumed to be an isotropic material. The constitutive equations can be expressed as [52], (3.3) ee ~ \2^rr C + \ fiee C C (3.4) 12^zz 42 (3.5) (3.6) Fig. 3.5 Geometry of a unit cell of a 1-3 piezocomposite and coordinate system, where c , c u and 12 are elastic constants and =(c -c )/2 . n 12 Substitution of eqns (3.3) - (3.6) and the strain-displacement relations in eqn (2.8) in eqn (2.6) yields the following governing equations expressed in terms of the displacements u and u . r 2 r^2 du ldu r+—Adr r dr V 2 r r 2 r^2 3 « 2 '44 dr —H 2 1 TU ,2 r du 2 r 2 dz drdz 2 1 oV du r dr oz - \d u ( 2 z ( =0 {d u 1 d^_ \drdz dz 2 r =0 A displacement potential function <p(r,z) is introduced such that d(p " "a7 r= dcp ~dz where p is a constant to be determined. Uz=p 43 (3.7) (3-8) Substitution of eqn (3.9) in eqns (3.7) and (3.8) leads to the following set of governing equations expressed in terms of <p{r,z). r^2. (3.10) ( [c^p + {d <p 1 d(p c +c \-^- -+-— + C P^T = 0 az 2 n u 2 U (3.11) Following the analysis presented in Chapter 2, coordinates r and z and the displacements u and u are nondimensionalized by the outer radius b. Stress and elastic r 2 constants are nondimensionalized by c^. For convenience, the nondimensional coordinates, displacements, stresses and elastic constants are denoted by the same symbols. In addition, the following nondimensional potential function is introduced. (3.12) b < P = \ 22 Using the nondimensional quantities, eqns (3.9) - (3.11) are changed to dm u, =^r-; dr ' d <p ( 2 { dr dm u -pdz (3.13) z i (3.14) r dr 2 7t\ d (p ]_d(p d <p =0 + cp dr dr " dz 2 (p+c +l) + l2 2 ny 2 (3.15) A nontrivial solution of eqns (3.14) and (3.15) exists if and only if, l + (l + c, )/?_ 2 c„ cp u (3.16) •=v p + c +\ l2 where v is a constant to be determined. Following quadratic equation of v is obtained by eliminating p from eqn (3.16). 44 77,v +77 v + 77 =0 (3.17) 2 2 3 where 77, (i = 1,2,3) are constants expressed in terms of material properties and are defined by eqn (A 12) in the Appendix. The two roots of eqn (3.17) are denoted by v,. (/ = 1,2), and for each root of eqn (3.17), eqn (3.16) yields a solution for p. It is convenient to denote the corresponding solutions by p (i = 1,2), where i identifies the corresponding root v,.. i The potential functions <p. (i' = l,2) corresponding to the roots v. (i = l,2) are ( governed by, ) (5 1 dm. 2 dc 2 8—f—i— _I i ^— _ + v.3— E-_ _ i + dr + V l r dr 2 (, = ,,2) 0 (3.18) ' dz' The displacements can be rewritten in terms of <p as t +¥>)•> dr u 2 = P l ^ dz + p ^ dz 2 (3.19) The solutions for stresses in terms of <p are given by eqns (A 13) - (A 16) in t Appendix. The acceptable form of the general solution of eqn (3.18) for composite cylinder problems can be expressed as, ft = [G 70^)+!,A:„te)][?«w(^)+J?««(^)] ( (3.20) where £, G L , P and R (i = 1,2) are arbitrary functions to be determined. n t The potential function should contain only even functions of z due to symmetry of the problem about z = 0. The solution for 7p can be expressed as, 45 <p = Z(r -2z]% +B Inr + y 2 0L ;=i j^ tkiCJiK ;=i «=i,3,5 + K (£ r )L ]cos{C„z) 0 n i in (3-21) where r^r^; z,=z/^; C =^ AQ,, G, , L, (i = 1,2) and B n n (3-22) n 0L are arbitrary functions to be determined from the boundary conditions. Because the normal pressure applied on the top and bottom surfaces of a composite cylinder cannot be expanded into a single functional variation of the radial v coordinate, it is prudent to make £"„ = so that the contribution from the series part of 2h eqn (3.21) to the vertical stress would be zero at the two ends where z = ±h. Substitution of eqn (3.21) in eqns (3.19) yields the following expressions for displacements. «,M = Z I C V ^ k ^ ^ R - ^ l C ^ K l ^ ^ C ^ + s t ^ r + ^ l ;=1 i'=l n=l,3,5 uAr.zh-tPt i=l tdhiCnW, +K {£s pJpirt£ z)-4Yp &Lz 9 l «=1,3.5 K i=l (3.23) r l (3.24) ^. 3.3 Piezocomposite Cylinder Subjected to Hydrostatic Loading In this section, the piezocomposite cylinder shown in Fig. 3.5 subjected to hydrostatic pressure q on all sides is analyzed by using the general solutions derived in 0 the preceding section. Superscripts / and m are used to denote the quantities associated with the piezoelectric fiber and polymer matrix respectively. The piezoelectric rod is perfectly bonded to the surrounding polymer matrix along the interface and the rod is electrically insulated along the interface. The boundary conditions for a unit cell subjected to hydrostatic pressure loading can be expressed as 46 o{ {r,±h) = -q ,D (r,±h) = 0,ol{r,±h) = 0 for 0<r<a (3.25) O (r,±h) = -q ,O {r,±h) =0 a<r<\ (3.26) f 2 0 2 n r, 2 2 Q 2 r for O? {\,z) = -q , < ( U ) = 0 r for 0 D {a,z) = 0 for f r (3.27) -h<z<h (3.28) -h<z<h Along the fiber-matrix interface, the continuity of displacements and tractions (perfect bonding) is required. Therefore, a f rr {a,z) = ol {a,z); u{ {a,z) = ol {a,z)=a (a,z); (3.29) {a,z) m n (3.30) u{ (a,z) = < (a,z) By using the eqns (2.26), (2.27) and (A2) - (A5), the general solution for the piezoelectric rod can be expressed as, £ oo 3 ul(r,z)= i=l «! M = . 3 (3.31) Ys i» n^i i( i) (s„z)+^A r G s I s r cos n 0i i=l n=\,3,5 3 tfi^nh (V/ )sin{s z)-Af k n 1=1 j 1=1 n=l,3,5 1=1 n=l,3,5 li (3.32) ^z ">i L cos V 1=1 2Z(c/;+c -2 .k / 1 2 Z (3.33) ; 0 « M = X E i* «4lfe"M, KVVofal)+ fe - 4 ) ~ A ) cos G s i=l n=l,3,5 L r • . 2Zfe+c -2//,K,. • / 1 2 1=1 (3.34) (3.35) i=l n=l,3,5 1=1 47 (3.36) i=l n=l,3,5 I=1 n=l,3,5 YPM A (VI Df {r,z) = (3.37) W 3 i=l I n=l,3,5 (3.38) /=1 By using the eqns (3.19) and (A13) - (A15), the general solution for the polymer matrix annular cylinder can be expressed as, 3 <(r,z) ~ -, r = £ 5XV^[/,(Cn)G in x in _ 2 -K (Cs)L ]cos(£ z)+ n 2£4 r+B J w m + ^ , ^ ) T i « ^ ) - 4 & ^ L * /=1 i=l n=l,3,5 «."M = - Z P i t^ih^M,, ± r (3.39) (3.40) i-=i >cos i=l n=l,3.5 (cu +Zi fc.VM^ofe)-te - ^ ^ . f e ) r (3.41) >cos 1=1 n=1.3.5 (3.42) (3.43) /=1 n=l,3,5 1=1 n=l,3,5 i'=l (3.44) 48 Note that all stresses and elastic constants of the piezoelectric and elastic material are nondimensionalized by c£ and displacements and coordinates are nondimensionalized by the outer radius of the cylinder (b) which is set as the nondimensional unit length parameter. Because 0"™(l,z) = O and a" (l,z) = -q , r the eqns (3.44) and (3.41) can be 0 expressed as, £ 5 CAlXC bfc -KXLbfi ]=Q n n (3.45) in =0 iflfu+c^-lZ )AOI: +(cr - c ^ K , 2 = -q 0 (3.46) (3.47) i=i In view of D{{a,z) = 0, the eqn (3.37) can be expressed as, (3.48) i=i Following relationship between G,„ and L (i = 1,2) can be obtained from eqns in (3.45) and (3.46). G\„- T,„L K ln (3.49) + K „L 4 2n (3.50) G =-fcnL +K L „) 2n ]n 2n 2 Using the eqn (3.48), Q _ ^2nG +^3„G „ 2n (3.51) 3 0),In where K (i = 1,...,4) and co (/ = 1,2,3) are listed in the Appendix [(A20) - (A21)]. in in 49 At the two ends of the cylinder, the boundary conditions are cr{ (r,±h) = -q , 2 0 D{ (r,±h) = 0 and <7™(r,±/i ) = - # „ . The trigonometric series part in the general solution of a{ , D{ and <T z are all equal to zero when z = ±h. Therefore, using eqns (3.35), 22 (3.38) and (3.43) with z = ±h, - 4 l > 4,,. (3.52) (=1 & 4 = 0 (3.53) AJjJ^^-q, (3.54) W Assuming that the trigonometric series converge for N terms, there are AN unknowns involving G , G , L 2n 3n u and L 2n and 6 unknowns involving the constant terms, AQ, (/ = 1,2,3), A (/ = 1,2) and B . 0I 0L Now consider the continuity conditions at the interface. Substitution of eqns (3.33) and (3.41) into the condition (j (a,z) = <7™(a,z) leads to f 3 2 ZA G,,=XkG,,+^,lJ /•=i 2Efe 1=1 (3.55) n 1=1 +c{ -2 2 Xl K , =2X(cn +c™-2Z i=i ) \ +( C l Substitution of eqns (3.36) and (3.44) in the condition --c^K (3-56) a <J ( > ) zA > ) f z r a z =<J a z leads to the following relationship. I W i v , ^ =tfefeR-^vMC^)I i=i i=i i n ] (3.57) Eqns (3.31) and (3.39) together with the interface condition u{(a,z) = u™(a,z) yield, 50 XVV,(^,X =t f e f e X /=i -fiiKtijafo] (3.58) 1=1 2 ^ 0 =2 ^ 0 + - ^ £f 7=T (3.59) « Now only the continuity condition u{(a,z) = u "(a,z) remains to be satisfied. In z the interface, r remains constant and z varies. Eqns (3.32) and (3.40) have a trigonometric series and a linear z term. The linear term of z has to be expressed in terms of a trigonometric series of sin(£ z) n before applying the interface continuity condition. Therefore, the following Fourier series expansion is introduced. z=t^^sin(C z) (3.60) n n=i.3.5 " n Substitution of eqn (3.60) in eqns (3.32) and (3.40) results in expressions of identical variations with respect to the vertical coordinate z. Thereafter using the vertical displacement continuity at the interface yields the following relationship. XsA/ok«,)G,„ -W* ^iknPJoiLaMn +LpMLa )L ]-W: <=i i i=i in (3.61) Note that f3 (/ = 1,2,3), rj , 0) (i = \,2), W and W are presented in the f in in in n m n Appendix [(A22) - (A25)]. Eqns (3.55), (3.57), (3.58) and (3.61) represent 4N linear algebraic equations and eqns (3.47), (3.52), (3.53), (3.54), (3.56) and (3.59) form the remaining 6 equations to solve for the (4N + 6) unknowns. Consider a piezoelectric element subjected to hydrostatic pressure denoted by q . 0 Using the basic constitutive equation, D =d Mrr z 3 +<rj+d 3<7:: 3 + 4 ^ = d„{-q )+£°E 0 z (3.62) where d is the hydrostatic piezoelectric coefficient and the superscript a means under h constant stress. 51 The stress-induced electric displacement D in eqn (3.62) consists of three parts: z a component due to lateral stresses, a component due to axial stress and another due to vertical electric field. In the case of a unit cell made only of a piezoelectric rod subjected to hydrostatic pressure, the lateral stresses and vertical stress would be equal to the applied hydrostatic pressure and the hydrostatic piezoelectric coefficient, d =2d + d . h M 33 Therefore, d of a monolithic piezoelectric unit cell is low due to the opposite signs of h the piezoelectric coefficients, Sh ~dh/ 33 e d n and d . The hydrostatic voltage 3l coefficient, > also becomes very small because of the relatively large magnitude of the dielectric constant e . For piezocomposites, the eqn (3.62) is written in terms of the 33 volume averages of the stresses, electric displacement and electric field. Therefore, (D ) = {d a ) + {d a ) + {d a ) + ( ^ z 3l rr 3l ee 33 zz (3.63) £ and d= 1 X (3.64) 1 h where ( ) represents volume average and d is the effective hydrostatic piezoelectric h coefficient. For example, the volume average of d a 3] is defined as, rr lid^JdV ( d ^ r r ) = V (3-65) / / m where V and V denote the volume of the fiber and matrix phase of a unit cell f m respectively. The effective hydrostatic voltage coefficient, g , is calculated from, h 52 Because of the stress transfer from the passive polymer phase to the piezoelectrically active ceramic phase, the hydrostatic response of a piezocomposite can be dramatically improved when compared to a monolithic piezoceramic element. In the next chapter dealing with numerical results, the influence of matrix and piezoelectric rod properties, rod aspect ratio and rod volume fraction on the hydrostatic performance of a 1-3 piezocomposite is investigated by evaluating the effective hydrostatic piezoelectric coefficient d and voltage coefficient h g. h 53 Chapter 4 NUMERICAL RESULTS AND DISCUSSION Electroelastic field in solid and annular piezoelectric cylinders and a 1-3 piezocomposite unit cell is investigated in this chapter by numerical implementation of the analytical solutions derived in Chapters 2 and 3. The main objective is to examine the influence of material properties, geometry and different loading conditions on the behaviour of cylindrical actuators and a piezocomposite unit cell. Examination of the characteristic features of the electroelastic field in cylinders would enhance the current understanding of behaviour of actuators and 1-3 piezocomposites. The first step of the numerical analysis is to establish the accuracy of the current numerical solutions by comparing with selected solutions available in the literature. There are not any solutions reported in the literature for piezoelectric cylinders. The available solutions for elastic cylinders are therefore used to establish the accuracy of the piezoelectric solutions by setting the piezoelectric and dielectric constants to negligibly small values. 4.1 Comparison with Elastic Cylinders Vendhan and Archer [24] analyzed a transversely isotropic elastic solid cylinder subjected to various boundary loads. One of the loading cases considered by them corresponds to the following boundary conditions for a solid cylinder of radius b and height 2h. (T =0, <T„=0, rr at r = b (4.1) ^ ? W ^ ° . at z = ±h = (4.2) where q(r)=l—2r = 2 . Before proceeding any further with calculations, it is important to examine the convergence of the loading function series expansion given by eqn (2.91). In the case of a 54 solid cylinder, a is equal to zero and H (t r) is simplified to J {t r), where t 0 m 0 m m is the w-th root of 7, (t r). Therefore, using eqn (2.91) m l{r) = l-2r =Q + 2 0 (4.3) f Q J {Lr) 4 m 0 m=l where (4.4) Q =2[q{r)rdr = Q 0 8 _ {qi&J&mrty [rJl{t r)dr t m 2 m J 0 (4.5) { t m ) Result from eqn (4.3) Result from q(r)=1-2r CT 2 o.o \- 0.0 0.2 0.4 0.6 r Fig. 4.1 Comparison of series expansion of q(r)=l-2r . 2 Fig. 4.1 shows a comparison of the series expansion for M = 30 and it is noted that the series expansion accurately simulates the applied normal pressure and convergence is reached for M > 25. Next, consider the case of a cylinder subjected to uniform normal pressure over a part of the top and bottom surfaces, i.e. j-q 0 (0<r<b ) ^ 0 (4.6) 55 Taking b =0.56, q(r) can be expressed using eqn (4.3) with 0 Qo= l o rdr = Ss. 4 (4.7) 2 0 [\«rJ {t r)dr Q Q m = m [rJl(t r)dr m _ ^ V (4.8) ^ J tJo(0 200 400 600 800 1000 M Fig. 4.2 Series expansion of the discontinuously applied loading at r=0 for different M values. Fig. 4.2 shows the solution of q(r) at r = 0 from the series expansion for different values of M. Convergence is very different for this discontinuous loading function when compared to Fig. 4.1 where convergence and accurate simulation of the loading function were achieved for M > 3 0 . It appears that even with a very large number of terms (M>600), the loading function has a small error (2%) at r = 0. To further understand the convergence and behaviour of the series expansion of the loading function, Fig. 4.3 presents q(r) at r = 0.0056 for different M and Fig. 4.4 shows q(r) calculated from the eqn (4.3) over the range 0<r<b for M = 150. It is clear from Fig. 4.2 - 4.4 that eqn (4.3) has convergence problems only near r = 0 and 0.56 for the discontinuous loading given by eqn (4.6). Fig. 4.4 shows that the series expansion is accurate within 1 % of the exact value at other values of r. 56 In addition to the examination of stability and convergence of the Bessel series expansion of the applied loading, it is necessary to examine the convergence of the series solution of the electroelastic field as expressed by eqns (2.34) - (2.53). Note that in Chapter 2 various quantities are made nondimensional to ensure overall numerical 57 stability of the solution. It is observed from eqns (2.34) - (2.53) that another source of numerical instability is associated with the hyperbolic sine and cosine functions and the modified Bessel functions with increasing number of terms (m and n indices) of the series solution. To prevent such numerical instability, all terms involving the summation indices n and m are normalized with respect to I (s„b^) and cosh(t hi) respectively. x m Fig. 4.5 shows the nondimensional normal stress, <?* =O' /q , at r = 0.0056, zz zz 0 0.4956 and 0.756 of a solid cylinder (h/b = \) subjected to boundary conditions given by eqns (4.1) and (4.6) and computed by using the analysis presented in Chapter 2. The results corresponds to M = 150 (loading series expansion) and different values of N [series expansion in eqns (2.46) - (2.53)]. Fig. 4.5 confirms that the numerical results for vertical stress at r = 0.0056 and r = 0.756 are stable with increasing N and agree with the boundary condition at the top surface within 1 % accuracy. o.o k -0.2 h -0.4 h -0.6 k - r=0.005b r=0.495b r=0.750b -0.8 h -1.0 50 60 70 80 90 _i_ 100 110 _i_ 120 130 140 150 N Fig. 4.5 Nondimensional vertical stress at selected points at z'-l of a solid cylinder for different values of N. Fig. 4.6 shows a comparison of vertical stress profile of a solid cylinder (h/b = l) along r = 0.0056 for M = 150 and for five different values of T V . It is clear that the numerical solution is stable and convergent. A comparison of vertical stress profiles at r = 0 and 0.0056 also confirms that slow convergence is only limited to the center of the 58 top and bottom surfaces of the cylinder. Based on the numerical results presented in Fig. 4.1 - 4.6, it can be concluded that M,N> 30 is sufficient for stable and converged solutions for solid cylinders subjected to smooth loading functions similar to eqn (4.2). For discontinuous loading functions of the form of eqn (4.6), stable and convergent solutions require M>150 and N>50 to achieve high accuracy of solutions in the vicinity of discontinuities of the loading function. -0.3 -0.4 -0.5 -0.6 -0.8 -0.9 -1.0 -M=150N=150 (r=0) -1.1 0.0 0.2 0.6 0.4 0.8 1.0 z Fig. 4.6 Nondimensional vertical stress profiles along z-axis (r=0.005b) of a solid cylinder (h/b=l) for different values of M and N. After confirming the stability and convergence of the Bessel series expansion of the applied loading and the series solution of the electroelastic field (Chapter 2), attention is now focused on the special case of a solid elastic cylinder considered by Vendhan and Archer [24]. They considered a magnesium solid cylinder subjected to the boundary condition given by eqns (4.1) and (4.2). The properties of magnesium are C =5.64, n C =5.86, C =2.30, C =1.81, ^=1.68 33 12 13 59 (x\0 Nm- ). ]0 2 + A r=0 (Present) r=0 (Ref. 24) r=0.5b (Present) r=0.5b (Ref. 24) / < - 0.4 - , / / A Ass^-A-^O ' I . I . 0.0 0.2 0.4 0.6 0.8 1.0 Z (a) Nondimensional hoop stress profiles along z-axis (h/b=l) 0.10 aja -0.15 - + aJo oja -0.20 - A -0.25 l 0.0 • (Present) \ (Ref. 24) (Present) ojo' (Ref. 24) 1 ' 1 0.2 • 0.4 1 0.6 • 1 0.8 • ' 1.0 r (b) Nondimensional radial and hoop stress profiles along r-axis (z=0, h/b=0.2) Fig. 4.7 Comparison of solutions for stresses of a solid magnesium cylinder. Fig. 4.7 shows a comparison of present solutions for radial and hoop stresses of a magnesium cylinder with those obtained by Vendhan and Archer [24], where cr* =<jr(o) = l . In Fig. 4.7 (a), hoop stress variation along the vertical axis (z* =z/b) at 60 r = 0 and r = 0.56, is compared for a cylinder with h/b = 1.0. In Fig. 4.7 (b), both radial and hoop stress profiles along the r-axis (r* =r/b) at z = 0 are compared for a cylinder with h/b = 0.2. The two solutions agree very closely and confirm the high accuracy of the present numerical results. Okamura [26] analyzed the case of a transversely isotropic elastic hollow cylinder subjected to an outer band of load by using analytical techniques similar to the ones used in this thesis. The geometry of the hollow cylinder is such that b/a = 4, b/h = l and d/h = 0.3 (Fig. 2.1) and d denotes the half-width of the band load. A magnesium cylinder was used in the numerical study. The boundary conditions used by Okamura [26] for an elastic cylinder were: a =0, a =0, rr n { - <7 n 0 <7 =0, zz n at r = a (4.9) Izl < d k / |z| > d < T -= ' 0 cr ,=0, z a t r = b at z = ±h ( 4 - 1 0 ) (4.11) Analysis of a hollow cylinder is computationally more laborious than a solid cylinder due to the additional boundary conditions involved and the numerical solution may be susceptible for severe ill-conditioning when the ratio of the inner radius to outer radius approaches unity. Numerical solutions for elastic hollow cylinders are found to converge for M = 11 and N = 11 for the loading conditions considered by Okumura [26]. Fig. 4.8 shows a comparison of current results with those obtained by Okumura [26]. The two solutions agree very closely. Note that vertical stress at z/h = \.0 has to be equal to zero because of the boundary condition but the numerical solution is not exactly zero at these locations due to minor precision errors. 61 0.2 0.0 o .,--- f ......cr D K -0.2 , cr O -0.4 A;- -0.6 ,7 / / / ...A"'' , -0.8 ° + A o d • r=0.40b (Present) =0.40b (Ref. 26) r=0.55b (Present) r=0.55b (Ref. 26) r=0.70b (Present) r=0.70b (Ref. 26) r=0.85b (Present) r=0.85b (Ref. 26) r J3-"'' -1.0 i 0.0 i I 0.2 I i 0.4 I i 0.6 I i 0.8 1.0 Z (a) Nondimensional radial stress profiles along z-axis 0.4 r=0.40b (Present) r=0.40b (Ref. 26) r=0.55b (Present) A r=0.55b (Ref. 26) r=0.70b (Present) O r=0.70b (Ref. 26) r=0.85b (Present) • r=0.85b (Ref. 26) + 0.3 0.2 5" 8 0.1 F "A 0.0 -0.1 -0.2 0.0 ..-a —0.2 ..o .-6 0.4 0.6 0.8 1.0 (b) Nondimensional vertical stress profiles along z-axis Fig. 4.8 Comparison of stresses of a hollow magnesium cylinder. 62 4.2 Electroelastic Field of a Solid Cylinder under Mechanical Loading In this section, the basic features of coupled electroelastic field of a solid piezoelectric cylinder are examined. Three piezoelectric materials and different aspect ratios of cylinder (i.e. h/b = 0.5,0.75,1.0,2.0,5.0) are considered in the numerical study. Table 4.1 shows the material properties of the three piezoelectric materials, i.e. BaTi0 , 3 PZT-4 and PZT-5H. Table 4.1 Material properties of selected piezoceramics. PZT-4 PZT-5H 15.0 13.9 12.6 c (io W ) 14.6 11.5 11.7 c (lO Nm- ) 6.6 7.78 7.95 c (l0 Nm- ) 6.6 7.43 8.41 '44 (l0 Nm- ) 4.4 2.56 2.3 (Cm' ) 11.4 12.7 17.0 '31 {Cm- ) -4.35 -5.2 -6.55 '33 (Cm- ) 17.5 15.1 23.3 (lO-'Fm- ) 9.87 6.45 15.38 (10-V//T') 11.15 5.62 12.76 BaTiO, c„ (lO Nm- ) w 2 I( 2 33 w l2 2 l0 u 2 l0 2 2 2 2 1 £33 Consider the case of a solid cylinder which is subjected to uniform vertical pressure q on the top and bottom ends as shown in Fig. 4.9. The boundary conditions 0 are expressed by eqns (4.1) and (4.6). In addition, D (b,z) = 0 and D (r,±h) = 0. r z 9o h o Fig. 4.9 Piezoelectric cylinder subjected to uniform vertical pressure. 63 -0.10 BaTiO, -0.50 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 4.10 Nondimensional vertical stress profiles along the r-axis (z=0). 0.13 BaTi0 PZT-4 PZT-5H 0.12 3 0.11 0.10 0.09 0.08 0.07 0.06 0.05 I 1 0.0 0.2 0.4 . I 0.6 . 0.8 1.0 r Fig. 4.11 Nondimensional hoop stress profiles along the r-axis (z=0). Fig. 4.10 and 4.11 show the nondimensional vertical stress, cr =cr /^ , and zz zz 0 hoop stress, o" = CTgg/q , profiles of a cylinder (h/b = l, b /b = 0.5) along the radial axis ee 0 0 at z = 0. Vertical stress is compressive but hoop stress is tensile at the middle of the cylinder. Vertical stress in BaTi0 3 is slightly higher than that in PZT materials. This 64 trend is reversed in the case of hoop stress and the maximum hoop stress in the cylinder mid-plane is lower than 15% of the applied pressure. Fig. 4.12 shows the variation of nondimensional vertical electric field, E* =E e Jq , z z i along the radial axis at z = 0. 0 Vertical electric field in PZT materials is more than twice that in BaTi0 and the 3 variation with the radial coordinate is qualitatively similar for all three materials. -0.01 -0.02 -0.03 -0.04 BaTi0 -0.05 3 PZT-4 PZT-5H I 1 0.0 0.2 0.4 0.6 . I . 0.8 1.0 r Fig. 4.12 Nondimensional vertical electric field profiles along the r-axis (z=0). -0.3 .11 I ' 0.0 i 1 0.2 i i 0.4 i 0.6 1 0.8 i I 1.0 z Fig. 4.13 Nondimensional vertical stress profiles along the z-axis (i=0). 65 BaTi0 PZT-4 P2T-5H 3 -0.05 -0.10 - -0.15 -0.20 1 0.0 . 0.2 1 0.4 0.6 0.8 1.0 z Fig. 4.14 Nondimensional vertical displacement profiles along the z-axis (r=0). -0.1 I 0.0 • i 0.2 • i 0.4 i i 0.6 i i 0.8 i I 1.0 z Fig. 4.15 Nondimensional vertical electric displacement profiles along the z-axis (r=0). Fig. 4.13 and 4.14 show the variation of nondimensional vertical stress and vertical displacement, u* =u c^/bq , z z Q with z*=z/b along the center of the cylinder (r = 0). Vertical stress rapidly decays along the z-axis and the magnitude at the center is less than 50% of the applied pressure. Minor numerical precision problems exist in the 66 vertical stress profiles at z* = 1 due to reasons explained earlier. Nondimensional vertical displacement increases in a near linear fashion along the length of the cylinder and this behaviour is similar to one-dimensional response of a long cylinder under uniform load. Vertical stress and displacement of a BaTi0 cylinder are slightly higher than those in 3 PZT cylinders. Fig. 4.15 and 4.16 show the variation of nondimensional vertical electric displacement, £) * =Z) c /e ,<7 , and vertical electric field along the z-axis at r = 0. The z z 44 3 0 vertical electric displacement should be zero at the top end of the cylinder but this boundary condition is not satisfied exactly due to precision errors. D] remains positive along the cylinder axis and increases rapidly from zero to its peak value near z* =0.7 and thereafter gradually decreases with z*. Vertical electric displacement in BaTi0 is 3 higher than that in PZT materials. Vertical electric field is negative along the axis of the cylinder and is non-zero at the top end. The peak value of E* occurs near z* = 0.7 and PZT materials generate a larger nondimensional vertical electric field in the central region of the cylinder when compared to BaTi0 . 3 -0.02 h -0.03 -0.04 HI -0.05 -0.06 - BaTi0 PZT-4 PZT-5H 3 -0.07 -0.08 0.0 0.2 0.4 0.6 0.8 1.0 z Fig. 4.16 Nondimensional vertical electric field profiles along the z-axis (r=0). 67 The influence of cylinder geometry {h/b) is examined by considering cylinders made of PZT-5H with different h/b ratios and b /b = 0.5. 0 0.2 -0.2 - -0.4 / -0.6 - h/b=0.5 h/b=0.75 / h/b=1 h/b=2 h/b=5 -0.8 -1.0 0.0 I 0.2 . I 0.4 . 0.6 I . 0.8 1.0 r Fig. 4.17 Nondimensional vertical stress profiles along the r-axis (z=0) of PZT-5H cylinders for different h/b ratios. 0.18 0.16 0.14 0.12 0.10 e s o.08 0.06 0.04 h/b=0.5 h/b=0.75 h/b=1 h/b=2 h/b=5 0.02 0.00 0.0 0.2 _i_ 0.4 0.6 0.8 1.0 r Fig. 4.18 Nondimensional hoop stress profiles along the r-axis (z=0) of PZT-5H cylinders for different h/b ratios. Fig. 4.17 and 4.18 show the nondimensional vertical stress and hoop stress along the r-axis at the center of a cylinder (z = 0). As expected the radial stress distribution 68 becomes uniform as h/b increases and it is clear that for h/b>2 the solution approaches that of a long cylinder subjected to uniform end loading. For example, the vertical stress is equal to 0.25 for h/b = 2 and 5 and this value is exactly equal to the 1-D solution for a long rod. Therefore three-dimensional stress analysis is needed only if h/b<2 . Fig. 4.18 shows an interesting behaviour of hoop stress where for very short cylinders (h/b = 0.5) the peak value is observed at the outer surface of the cylinder instead of the center of the cross-section. For h/b>2, hoop stress is nearly zero which is consistent with the 1-D theory. -0.02 -0.03 -0.04 -0.05 . N -0.06 LU -0.07 -0.08 -0.09 -0.10 0.0 0.2 0.4 0.6 0.8 1.0 r Fig. 4.19 Nondimensional vertical electric field profiles along the r-axis (z=0) of PZT-5H cylinders for different h/b ratios. Fig. 4.19 shows the variation of nondimensional vertical electric field along the raxis at the center of the cylinder (z = 0). The radial variation is qualitatively similar to that of vertical stress and the magnitude of E* approaches the 1-D solution of 0.0323 for z h/b>2. Fig. 4.20 shows the variation of nondimensional vertical stress along the cylinder axis at r = 0 for cylinders of different h/b ratios. As noted previously, for relatively long cylinders (h/b>2), vertical stress decays rapidly from the loaded ends and approaches 69 the 1-D solution in the middle part of the cylinder. In the case of short cylinders, the decay of vertical stress with distance from the top end is significant but not as high as in the case of long cylinders. For a short cylinder (h/b = 0.5), in order to achieve good results on the top surfaces, a large value for M is needed. Here M is chosen to be 353. -0.2 -0.3 ""'X - \ \ -0.4 -0.5 \ \ \ \ \ \ - \ \ \ \ \ \ -0.6 O -\ \ \ \ \ -0.7 -0.8 -0.9 -1.0 \ \ \ \ \ \ \ \ h/b=0.5 h/b=0.75 h/b=1 h/b=2 h/b=5 \ I \ \ \ \ -1.1 0.0 \ ! \ - \ I \ \ •' 0.5 \ 1.0 1.5 2.0 2.5 3.0 3.5 4.0 \ \ \ \ \ 4.5 5.0 Fig. 4.20 Nondimensional vertical stress profiles along the z-axis (r=0) of PZT-5H cylinders for different h/b ratios. 0.7 — h/b=0.5 h/b=0.75 h/b=1 — h/b=2 - - h/b=5 0.6 0.5 /\ 0.4 0.3 0.2 0.1 0.0 0.0 _!_ 0.5 _!_ 1.0 _1_ 1.5 _1_ 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Fig. 4.21 Nondimensional vertical electric displacement profiles along the z-axis (r=0) of PZT-5H cylinders for different h/b ratios. 70 Fig. 4.21 and 4.22 show the variation of the nondimensional electric displacement and electric field with z*=z/b at r = 0 for cylinders of different h/b ratios. These profiles confirm that maximum electric displacement and electric field occur inside the cylinder near the loaded end except in the case of h/b = 0.5 . The maximum value of E\ is substantially larger than the value at the loaded end or the middle of the cylinder and same is true for the electric displacement. However, in the case of very short cylinders (e.g. h/b = 0.5) both D* and E* increases rapidly with depth and the peak values occur z z at the middle of a cylinder. -0.02 -0.03 -0.04 -0.05 'uj N -0.06 -0.07 -0.08 -0.09 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Z Fig. 4.22 Nondimensional vertical electric field profiles along the z-axis (r=0) of PZT-5H cylinders for different h/b ratios. 4.3 Electroelastic Field of a Solid Cylinder under Electrical Loading After examining the basic features of electroealstic field in a solid cylinder under mechanical loading, attention is now focused on a solid cylinder which is subjected to uniform electric displacement D on the top and bottom surfaces as shown in Fig. 4.23. 0 The boundary conditions are expressed by eqns (4.1), (4.12) and (4.13). 71 Fig. 4.23 Piezoelectric finite cylinder subjected to electric charge at the top and bottom ends. , < D = ° f-ZX (0<r<b ) x W o = D=0 i 0 fc< <6) < 7 r at r = b and -' = 0 -^ ° = * -h<z<h Z = ± h (4.12) (4.13) where b =0.56. 0 D(r) can be expressed in the form of eqn (2.106) with D =2^D rdr 0 = 0 D (4.14) n - _fV-/,(r,r)dr^ o D [rJt{t r)dr m y (4.15) tJl{t ) m The same values of M and N as in section 4.1 are used to obtain the numerical results. 72 0.004 0.002 h o.ooo F -0.002 -0.004 Fig. 4.24 Nondimensional vertical stress profiles along the r-axis (z=0) under applied electric charge loading. 0.003 0.002 0.001 0.000 h -0.001 h -0.002 \- -0.003 \0.4 0.6 1.0 r Fig. 4.25 Nondimensional hoop stress profiles along the r-axis (z=0) under applied electric charge loading. Fig. 4.24 and 4.25 show the variation of the nondimensional vertical stress, a zz = c r e / c £ ) , and hoop stress, &lg=<y e Jc D , along the radial axis at z = 0 of 22 31 44 0 0d 3 A4 o a cylinder with h/b = l. For PZT cylinders, the maximum value of cr* zz 73 occurs at the center and the magnitude decreases with the radial distance. Vertical stress experiences a change in sign in the vicinity of r = 0.65 for PZT cylinders and both compressive and tensile stresses exist at the middle of the cylinder. In the case of a BaTi0 cylinder, a\ 3 z remains tensile but relatively smaller in magnitude and nearly constant in the middle of the cylinder. Fig. 4.25 shows that hoop stress is tensile with its maximum value at the centre of the cylinder and then becomes compressive for r>0.5. Hoop stress in a BaTi0 cylinder is smaller than that in a PZT cylinder. 3 -0.008 BaTi0 3 PZT-4 -0.012 PZT-5H -0.016 * N LU -0.020 -0.024 -0.028 0.0 0.2 0.4 r 0.6 0.8 1.0 Fig. 4.26 Nondimensional vertical electricfieldprofiles along the r-axis (z=0) under applied electric charge loading. Fig. 4.26 shows the variation of nondimensional electric field, E* = z E e /c D , z 3l 44 0 along the r-axis at z = 0. The profiles are almost parallel to each other with a minor decrease in magnitude with the radial distance and the largest nondimensional electric field is generated in PZT-4 followed by PZT-5H and BaTi0 . Fig. 4.27 shows the 3 nondimensional vertical stress, o* , profiles along the z-axis at r = 0. Vertical stress zz should be zero at the top end of the cylinder, but this boundary condition is not exactly satisfied due to precision errors. Nondimensional vertical stress in BaTi0 cylinders due 3 to an applied charge remains nearly constant with depth and much smaller in magnitude 74 when compared to the stress generated in PZT cylinders. Maximum value of a\ occurs z in the vicinity of z* =0.5. It is interesting to note that o* zz of a PZT-5H cylinder is compressive along the length whereas nondimensional vertical stress corresponding to other materials is tensile. 0.006 0.004 0.002 0.000 -0.002 -0.004 -0.006 Fig. 4.27 Nondimensional vertical stress profiles along the z-axis (r=0) under applied electric charge loading. 0.07 Fig. 4.28 Nondimensional vertical displacement profiles along the z-axis (r=0) under applied electric charge loading. 75 "°- - BaTi0 8 3 PZT-4 •°- " PZT-5H 9 -1.0 i 0.0 i 0.2 i i 0.4 i i 0.6 i 0.8 i 1.0 Z Fig. 4.29 Nondimensional vertical electric displacement profiles along the z-axis (i^O) under applied electric charge loading. o.oo -0.02 -0.04 W N -0.06 -0.08 -0.10 0.0 0.2 0.4 0.6 0.8 1.0 z Fig. 4.30 Nondimensional vertical electricfieldprofiles along the z-axis (r=0) under applied electric charge loading. Fig. 4.28 u* =u e JbD , z i 0 and 4.29 show the nondimensional vertical displacement, and vertical electric displacement, D* =D /D , profiles along the z-axis z Z 0 at r = 0. The nondimensional vertical displacement increases with z* as in the case of a cylinder subjected to mechanical loading. PZT materials generate a larger overall vertical 76 displacement (stroke) when compared to BaTi0 . Vertical electric displacement decays 3 rapidly from the top surface of the cylinder and equal to -1 at z* = 1. D* of a BaTiO^ z cylinder is slightly higher than that of PZT cylinders. Fig. 4.30 shows the variation of nondimensional vertical electric field, E* , along the center of the cylinder (r = 0). z Vertical electric field is negative, decays slowly along the z-axis except near the ends and has its maximum value at the top surface of the cylinder. E* of a BaTiO z % cylinder is smaller when compared to that of PZT cylinders. 0.004 0.002 0.000 • „ -0.002 -0.004 -0.006 -0.008 0.0 0.2 0.4 0.6 0.8 1.0 r Fig. 4.31 Nondimensional vertical stress profiles along the r-axis (z=0) of PZT-5H cylinders under applied charge loading for different h/b ratios. Next, the influence of cylinder geometry {h/b) on the electroelastic field generated by electric loading is examined by considering cylinders made of PZT-5H with different h/b ratios. Fig. 4.31 shows the nondimensional vertical stress along the r-axis at z = 0 (mid-plane) for different h/b ratios. Nondimensional vertical stress has a large compressive value at the center of short cylinders and decreases with radial distance and becoming tensile close to the boundary of the cylinder. For a relatively long cylinder (h/b>2), <T* ZZ is nearly zero. For very short cylinders (h/b = 0.5), vertical stress at the mid-plane changes its sign twice and becomes compressive near the boundary. 77 0.020 h/b=0.5 h/b=0.75 h/b=1 h/b=2 h/b=5 1.0 Fig. 4.32 Nondimensional hoop stress profiles along the r-axis (z=0) of PZT-5H cylinders under applied charge loading for different h/b ratios. -0.15 -0.20 A - ^ -0.25 -0.30 -0.35 - / h/b=0.5 h/b=0.75 h/b=1 h/b=2 h/b=5 -0.40 -0.45 -0.50 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 4.33 Nondimensional vertical electric displacement profiles along the r-axis (z=0) of PZT-5H cylinders under applied charge loading for different h/b ratios. Fig. 4.32 and 4.33 show the nondimensional hoop stress and vertical electric displacement profiles along the radial axis at z = 0 for different h/b ratios. As h/b increases, hoop stress approaches zero and vertical electric displacement decreases rapidly. For a long cylinder (h/b>\), the radial distribution of D is nearly uniform. It is z 78 noted that vertical electric displacement is equal to -0.25 for h/b = 2 and 5 and this value is exactly equal to the 1-D solution for a long cylinder. -0.008 I— — —'— — — —'— —'— — — —•—— — — — — —'—i 1 1 0.0 1 0.5 1 1 1.0 1 1.5 2.0 1 1 2.5 1 1 3.0 3.5 1 1 1 1 4.0 4.5 5.0 Z Fig. 4.34 Nondimensional vertical stress profiles along the z-axis (r=0) of PZT-5H cylinders under applied charge loading for different h/b ratios. -0.2 \ \ - \ \ \ \ \ \ \ \ \ \ \ . \ \ -0.6 \ ! \ \ i -0.8 \ \ \ " \ \ \ \ \ \ \ \ i \ -1.0 \ \ I 0.0 0.5 1 1.0 . I h/b=0.5 h/b=0.75 h/b=1 h/b=2 h/b=5 \ I \ \ . 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Z Fig. 4.35 Nondimensional vertical electric displacement profiles along the z-axis (r=0) of PZT-5H cylinders under applied charge loading for different h/b ratios. 79 -0.01 -0.02 t" -0.08 • 0.0 1 1 0.5 1 1 1 1.0 1 1 1.5 1 1 1 2.0 1 1 2.5 ' 1 3.0 1 1 1 3.5 1 1 4.0 I 4.5 5.0 Z Fig. 4.36 Nondimensional vertical electric field profiles along the z-axis (r=0) of PZT-5H cylinders under applied charge loading for different h/b ratios. Fig. 4.34, 4.35 and 4.36 show the nondimensional vertical stress, electric displacement and electric field profiles along the center of cylinders of different h/b ratios. Maximum vertical stress occurs inside the cylinder near the loaded end except in the case of h/b = 0.5 and 0.75. For a long cylinder (e.g. h/b>l), the maximum value of a' zz is much larger than the value at the middle of the cylinder. For shorter cylinders (h/b<\.0), a\ decreases rapidly along the z-axis and the peak value occurs at the mid z plane of the cylinder. Vertical electric displacement decays rapidly with depthfromthe loaded end and approaches the 1-D solution for a long cylinder (e.g. h/b>2). For short cylinders, D* also decreasesfromthe top end but the value at the mid plane is larger than the 1-D solution. The variation of vertical electric field is nearly identical to the vertical electric displacement. 80 4.4 Electroelastic Field of an Annular Cylinder Mechanical Loading Consider an annular cylinder subjected to a uniform vertical pressure over a portion of the top and bottom surfaces as shown in Fig. 4.37. The boundary conditions corresponding to this problem are given by eqns (4.16) - (4.18). Fig. 4.37 Annular cylinder under vertical pressure. o = 0, rr D =0 at r=a (4.16) D =0 at r=b (4.17) tr =0, re cr =0, 0^=0, rr \-q 0 r r [a<r<b ) (4.18) 0 In the numerical study, b /b is set to 0.85 and different ratios of h/b (i.e. 0.5, 1, 0 2) and a/b (i.e. 0.25, 0.3, 0.5) are considered. The material of the cylinder is PZT-5H. Fig. 4.38 shows the nondimensional vertical stress, a] = CT /q , profiles of a cylinder 2 22 0 {h/b = \) along the vertical direction at r =0.7 (r* =r/b) for different a/b ratios. The solutions obtained from the commercial finite element package ANSYS are also presented for comparison. As expected, vertical stress is negative and decreases slowly from the top end to the middle of the cylinder where vertical stress is nearly constant. Fig. 81 4.39 shows the nondimensional vertical electric displacement, D* =D c /e q , z 2 44 il 0 in the z- direction at r = 0.7. The maximum value of D] is observed in the vicinity of z* = 0.84 for the three cases involving different a/b ratios. The peak value of D* increases as the z thickness of the cylinder decreases. -0.6 -0.7 * -0.8 -0.9 a/b=0.25 • a/b=0.3 a/b=0.5 a/b=0.5 (ANSYS) -1.0 0.2 o.o 0.4 0.6 0.8 1.0 z Fig. 4.38 Nondimensional vertical stress profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading. 0.3 a/b=0.25 a/b=0.3 a/b=0.5 a/b=0.5 (ANSYS) 0.2 0.1 \- 0.0 b-. -0.1 • <• Fig. 4.39 Nondimensional vertical electric displacement profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading. 82 -0.05 -0.06 -0.07 Hi" -0.08 -0.09 -0.10 0.0 0.2 0.4 z 0.6 0.8 1.0 Fig. 4.40 Nondimensional vertical electric field profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading. Fig. 4.40 shows the nondimensional vertical electric field E* =E e /q 2 2 3l 0 profiles in the vertical direction at r* =0.7. Vertical electric field has its minimum value at the surface, increases rapidly near the surface and becomes nearly constant with depth. E* 2 decreases with increasing a/b ratio. Vertical stress and vertical electric displacement profiles along the r-axis were computed at the center of the cylinder. It is found that vertical stress is nearly constant and vertical electric displacement is nearly zero for h/a>\. For short cylinders the radial distribution is non-uniform with peak values near the inside face of the cylinder. Fig. 4.41 shows the variation of nondimensional vertical stress with z at r* =0.7 for different h/b ratios. Similar to the case of a solid cylinder, for h/b>2 vertical stress decays rapidly from the loaded ends and approaches the 1-D solution in the middle part of the cylinder. Fig. 4.42 and 4.43 show the variations of nondimensional vertical electric displacement and electric field with z for different h/b ratios. D* reaches its maximum value near the loaded end of the cylinder and thereafter decreases rapidly with depth. Vertical electric displacement is nearly zero in the middle of the cylinder for h/b > 1. On 83 the other hand, vertical electric field is minimum at the loaded end, increases rapidly and thereafter show a gradual decrease with depth. Vertical electric field is negative at all points inside the cylinder. -0.60 - h/b=0.5 h/b=1 h/b=2 \ \ -1.05 1 0.0 0.5 • i 1.0 i i i 2.0 1.5 z Fig. 4.41 Nondimensional vertical stress profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading. 0.30 i 0.25 0.20 0.15 0.10 0.05 • h/b=0.5 h/b=1 h/b=2 0.00 -0.05 0.0 0.5 1.0 1.5 2.0 z Fig. 4.42 Nondimensional vertical electric displacement profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading. 84 • • - / / h/b=0.5 h/b=1 : h / b = 2 , / j I i i i i I j ... J...., J 0.0 0.2 i i 0.4 i i 0.6 • i 0.8 i • • i 1.0 1.2 1.4 1.6 1.8 2.0 z Fig. 4.43 Nondimensional vertical electric field profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under vertical loading. Electrical Loading Consider an annular cylinder subjected to uniform electric charge density D on 0 the top and bottom surfaces as shown in Fig. 4.44. The boundary conditions are given by eqns (4.16), (4.17) and (4.19). (4.19) Fig. 4.44 Piezoelectric annular cylinder subjected to electric charge loading. 85 0.0002 -0.0014 -0.0016 I 0.0 • 1 0.2 ' 1 1 0.4 1 0.6 1 L 0.8 1 1.0 z Fig. 4.45 Nondimensional vertical stress profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under electric charge loading. -0.6 , , -0.7 t -0.8 h -0.9 - a/b=0.25 a/b=0.3 a/b=0.5 -1.0 - 0.0 0.2 0.4 0.6 0.8 1.0 z Fig. 4.46 Nondimensional vertical electric displacement profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under electric charge loading. Fig. 4.45 shows the nondimensional vertical stress, <j* = cr e Jc D , zz zz 3 44 0 profiles in the z-direction at r* = 0.7 for three different values of a/b. Vertical stress is zero at the loading surface due to the boundary condition and increases rapidly reaching a maximum value in the vicinity of z* =0.8. Thereafter, vertical stress decreases rapidly with depth. 86 The magnitude of vertical stress increases with decreasing thickness of the annular cylinder. Fig. 4.46 and 4.47 show the variation of nondimensional vertical electric displacement, D* =D /D , z Z and vertical electric field, £* =is e /c .D , with z at 2 0 2 3 1 44 () r*=0.7. Vertical electric displacement has a unit value at the loading surface and decreases rapidly with depth near the top end. It is nearly constant within the middle half of the cylinder and the magnitude increases slightly with increasing thickness of the cylinder. The variation of vertical electric field with z is very similar to that of D* . The z distribution of vertical stress, vertical electric displacement and vertical electric field at the mid-plane of the cylinder is also considered in the numerical study but the numerical results are not given here. As in the case of solid cylinders, it is found that a , zz E\ are nearly constant at the mid-plane if h/b>\ cylinders (h/b<\) D* , and z and are non-uniform for shorter with peak values usually occurring near or at the inside surface of the cylinder. -0.03 • -0.04 - -0.06 \ - a/b=0.25 \ a/b=0.3 \ a/b=0.5 1 0.0 1 0.2 1 • 0.4 . 0.6 i . i 0.8 . 1.0 Fig. 4.47 Nondimensional vertical electric field profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under electric charge loading. 87 0.0002 0.0000 : i -0.0002 -0.0004 -0.0006 -0.0008 -0.0010 -0.0012 :\ I \ I -0.0016 -0.0018 0.0 h/b=0.5 h/b=1 h/b=2 \/ -0.0014 . i i i i 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 z Fig. 4.48 Nondimensional vertical stress profiles along the z-axis at r=0.7b of a PZTannular cylinder under electric charge loading. -0.6 Fig. 4.49 Nondimensional vertical electric displacement profiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under electric charge loading. 88 -0.035 -0.040 -0.045 -0.050 -0.055 - h/b=0.5 h/b=1 h/b=2 -0.060 -0.065 -0.070 0.0 1 ' 1 0.5 10 ' 1 1.5 ' 2.0 Z Fig. 4.50 Nondimensional vertical electricfieldprofiles along the z-axis at r=0.7b of a PZT-5H annular cylinder under electric charge loading. Fig. 4.48, 4.49 and 4.50 show the variation of nondimensional vertical stress, vertical electric displacement and electric field in the z-direction at r* = 0.7 for different h/b ratios. Vertical stress is zero at the top surface due to the boundary condition and increases rapidly with depth near the top surface and reduces to a negligible value in the middle of the cylinder for h/b>\. Similar to the solutions shown in Fig. 4.35 and 4.36, both of D\, and E\ decay rapidly with depth from the loaded end and approach the 1-D solution for a long cylinder in the middle part of the cylinder when h/b>2. 4.5 Response of a Unit Cell of a 1-3 Piezocomposite The response of a 1-3 piezocomposite unit cell under hydrostatic loading is considered in this section. According to author's knowledge, a three-dimensional analytical solution for a piezocomposite unit cell has not been reported previously. Given the complexity of the analytical formulation, it is prudent to compare the current numerical results with results obtained from another approach. Therefore, some results for a piezocomposite unit cell is obtained by using the commercial software code ANSYS. It is assumed that the piezoelectric core of a unit cell is made of PZT-5H and 89 three different isotropic elastic materials are considered for the polymer matrix. The properties of the matrix materials are given in Table 4.2. Table 4.2 Material properties of polymer phase of a 1-3 piezocomposite. Modulus (GPa) c 1 2 Case 1 Case 2 Case 3 14.135 2.827 0.565 6.055 1.211 0.242 0 -5 - ' •"""•"-i--.-v- --.i -10 - -15 - N -20 - - 2 5 Case 1 Case 2 Case 3 Case 2 (ANSYS) • -30 I 0.0 ' 1 0.2 • ' 1 0.4 1 0.6 ' 1 0.8 ' 1.0 r Fig. 4.51 Nondimensional vertical displacement profiles of the top surface (z=h) of a unit cell (a/b=l/6, h/b=5/3). u c p Fig. 4.51 shows the variation of nondimensional vertical displacement u] = b q z 44 0 at the top surface [z = h) with the nondimensional radial coordinate r' =r/b of a unit cell (a/b = 1/6 and h/b = 5/3) under hydrostatic pressure q 0 for different matrix materials. As expected, the vertical displacement of the ceramic rod is hardly influenced by the matrix material while vertical displacement of the matrix material is significantly influenced by its properties. Vertical displacement of the matrix is much larger than that of the piezoelectric rod and the maximum displacement is observed at the outer surface of a unit cell. The results obtained from ANSYS agree closely with the analytical solutions. 90 -5h -10 -15 - Case Case Case Case -20 \- 1 2 3 2 (ANSYS) _L_ -25 0.0 0.2 0.4 0.8 0.6 1.0 r" Fig. 4.52 Nondimensional radial displacement profiles at the centre of a unit cell (a/b=l/6, h/b=5/3). • - Case 1 Case 2 Case 3 Case 2 (ANSYS) • 1 0.2 o.o i i i i 0.4 i 0.6 i 0.8 i 1.0 r Fig. 4.53 Nondimensional vertical stress profiles at the centre of a unit cell (a/b=l/6, h/b=5/3). Fig. 4.52 * displacement u and 4.53 14 C show the radial variation of nondimensional radial * and vertical stress cr* = (T /q r z zz 0 at the centre of a unit cell for b q 0 different matrix materials. Radial displacement of the piezoelectric rod is negligible 91 (almost zero) but the radial displacement of the polymer matrix is quite close to its vertical displacement. This is reasonable as the matrix is isotropic and the strains in the radial and vertical directions would be similar due to hydrostatic loading. Finite element solutions for radial and vertical displacements agree closely with the analytical results. Fig. 4.53 shows that vertical stress of the piezoelectric rod is much larger than that of the elastic matrix, but the difference decreases as the stiffness of the matrix increases. Vertical stress is nearly constant within each phase and this is reasonable given that loading is hydrostatic and the unit cell is relatively long. The finite element results for vertical stress agree closely with the analytical solutions. 0.2 0.0 -0.2 -0.4 ^ -0.6 -0.8 -1.0 -1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 z Fig. 4.54 Nondimensional shear stress profiles along the interface of a unit cell (a/b=l/6, h/b=5/3). Fig. 4.54 shows the variation of nondimensional shear stress <7^ =<7 /<7 with the rz 0 vertical coordinate, z* =z/b at the material interface. Shear stress is zero at z" = 0 and increases with the distance from the centre. The dependence of interfacial shear stress on the matrix material is quite significant. Shear stress is singular near the material junction at r = a and z = h and the analytical results capture this behaviour and the finite element solutions fail to simulate the singular behaviour of interfacial shear stress at the material junction. However, the finite elements results agree within 5% of the analytical results at points away from the material junction. Interfacial shear stress profiles are important to 92 understand the load transfer mechanism between the two phases and potential for delamination failure near the material junction. Li and Sottas [35] studied the hydrostatic response of a unit cell of a 1-3 piezocomposite with a PZT-5H rod and an elastic matrix with Young's Modulus equal to 2.1 GPa and Poisson's ratio equal to 0.3. They used a unit cell with h/a = 13.3. The solution presented by Li and Sottas [35] is not based on the fully coupled piezoelectric governing equations considered in this study and is an approximate solution based on a uniform electric field in the rod. As mentioned in Chapter 3, the effective hydrostatic piezoelectric and voltage coefficients d and g of a unit cell are given by, h h (4.20) Fig. 4.55 and 4.56 show the variation of d and g as a function of the volume h h fraction of the piezoelectric rod in a unit cell. The present results are more reliable than the other two solutions because the plane strain model is unable to incorporate the influence of the finite rod aspect ratio and the effects of vertical pressure, and the approximate 3-D solution of Li and Sottas [35] is based on the assumptions that the electric field is constant in the piezoelectric rod and the mechanical and electric field are totally uncoupled. The 3-D model of the present study accounts for full electromechanical coupling in the rod and the 3-D elastic field in the matrix. However, the results obtained from the different methods show similar trends with the maximum value of d occurring at a volume fraction of 20-25% and the value of g peaking at a h h low volume fraction. The current result is very close to the solution by plane strain model for large h/a ratios, e.g. h/a > 13.3. But when h/a ratio decreases, result by the plane strain model is not accurate. In engineering applications, the h/a ratio typically varies from 7 to 3, therefore plane strain model cannot make good approximations in those cases. Furthermore, Li and Sottas [35] model has no electromechanical coupling in the ceramic rod and their results therefore do not account for the electric field generated in 93 the rod due to applied loading. All results converge to approximately similar limiting value as the volume fraction reaches 1.0. 2.0x10" Present (h/a=25) Present (h/a=13.3) Present (h/a=4) Plane Strain Model Ref. 35 (h/a=13.3) 2d +d 1.5x10"" s. ^X. //' /;/ / > 31 33 1.0x10" E ?' // -' c ./;// Ii / '' I 5.0x10" 11 • 'il . • f / 0.0 0.0 i 0.2 . i . 0.4 i 0.6 . i 0.8 1.0 V. Fig. 4.55 Effective hydrostatic piezoelectric constant d„asa function of volume fraction of piezoelectric phase. 0.20 0.16 — 0.12 E 0.08 0.04 0.00 Fig. 4.56 Effective hydrostatic voltage constant g„ as a function of volume fraction of piezoelectric phase. 94 Fig. 4.57 shows the variation of d with the piezoelectric rod aspect ratio (a/2h) h for a unit cell made out of PZT-5H or PZT-4 and the three different polymer materials defined in Table 4.2. The volume fraction of the piezoelectric rod is kept at 25% (a/b = \/2). The magnitude d increases as the aspect ratio decreases. Fig. 4.57 also h indicates that a softer matrix material can produce a higher effective hydrostatic piezoelectric constant. The effective hydrostatic piezoelectric coefficient of a unit cell made out of PZT-4 is substantially smaller than that of a unit cell made out of PZT-5H for low aspect ratios. 1.6x10"'° Case Case Case Case 1.4x10'° 1 2 3 2 (PZT-5H) (PZT-5H) (PZT-5H) (PZT-4) 1.2x10'° > i. -o 1.0x10 10 TJ 8.0x10" 6.0x10"" j i i i i i i i i i i i i i i i t 4.0x10"" 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 a/2h Fig. 4.57 Variation of d with piezoelectric rod aspect ratio (Vf=25%). h Fig. 4.58 and 4.59 show the variation of the effective hydrostatic piezoelectric constant d and the effective hydrostatic voltage constant g with the volumefractionof h h piezoelectric phase for a rod with aspect ratio, a/2h = 0.\. It can be seen from the Fig. 4.58 that the effective hydrostatic piezoelectric constant exceeds the hydrostatic piezoelectric constant of the constituent ceramic, reaching more than twice the value of pure PZT-5H (45xlO~'WK) and PZT-4 (43xlO" m/F) for all matrix materials 12 considered in this study and for volumefractionsgreater than 10%. The maximum value 95 of d occur between 20-40% volume fraction depending on the properties of the matrix h material. In contrast Fig. 4.59 shows that the maximum value of g occurs at low volume h fractions of the ceramic phase, attaining a value five to thirty times the value of i.4xi(r 1.2x10''° 1.0x10'° 8.0x10" 6.0x10" E^ T3 c Case Case Case Case 4.0x10" 2.0x10" 1 (PZT-5H) 2 (PZT-5H) 3 (PZT-5H) 2 (PZT-4) 0.0 -2.0x10" o.o 0.2 0.4 0.8 0.6 Fig. 4.58 Variation of d with volume fraction of piezoelectric phase for different matrix materials (a/2h=0.1). h 0.08 0.07 - Case Case Case Case 0.06 II \ 0.05 z E 0.04 1 (PZT-5H) 2 (PZT-5H) 3 (PZT-5H) 2 (PZT-4) \ \ i \ co 0.03 0.02 0.01 0.00 0.0 1 . i 0.2 0.4 » -i 0.6 i 0.8 Fig. 4.59 Variation of gh with volume fraction of piezoelectric phase for different matrix materials (a/2h=0.1). 96 pure ceramic [PZT-5H (L5xlO~ Vm/N) 3 and PZT-4 (3.8xlO" Fw/7V")]. This 3 enhancement is due to the dilution of the dielectric permittivity and the increase of d . h 3.2x10" 12 Case Case Case Case 2.8x10"' 2 2.4x10' 12 _ 2.0x10"' •§- 1.6x10"' Z 1 (PZT-5H) 2 (PZT-5H) 3 (PZT-5H) 2 (PZT-4) 2 2 1.2x10" 8.0x10'" 4.0x10'" 0.0 0.0 0.2 0.4 V 0.6 , 0.8 Fig. 4.60 Variation of d^g with volume fraction of piezoelectric phase for different matrix materials (a/2h=0.1). h Fig. 4.60 shows the variation of the figure of merit, d g , h of a 1-3 h piezocomposite as a function of the volume fraction of the ceramic phase. A peak is observed at a low volume fraction for essentially the same reason as the g peak. The h value attained is significantly higher than the value corresponding to a pure piezoceramic [PZT-5H-67.5xl0- m /JV and PZT-4 - 163.4xl0~ /n /N]. |5 2 15 2 . Ee Fig. 4.61 and 4.62 show the nondimensional vertical electric field E = and 2 31 z o a vertical displacement at the center of a piezocomposite unit cell (r = 0,z*=z/b) due to hydrostatic loading. It can be seen that the vertical electric field is not negligible along the z-axis. And when the a/h ratio is large, e.g. afh = \fl and 1/3, E* is also not 2 constant at the center of the unit cell. Vertical electric field along the z-axis is negative and the maximum value occurs at the mid-plane of the unit cell. When the a/h ratio is larger than 1/25, the maximum value of E\ decreases with increasing aspect ratio. The 97 results shown in Fig. 4.61 confirm that the approximation made by Li and Sottos [35] is not valid for both short and long unit cells. The nondimensional vertical displacement is negative and varies linearly along the z-axis except near the top and bottom surfaces of the unit cell. The maximum value occurs on the two ends and increases with decreasing aspect ratio. • - a/h=1/25 I / a/h=1/7 / j a/h=1/3 - J -0.08 -0.12 / - 0 / 2 4 6 8 10 12 14 Z Fig. 4.61 Nondimensional vertical electric field profiles along the z-axis of a unit cell (a/b=l/2). _4 0 I 0 i I 2 > i 4 • i 6 • i 8 • i 10 • i 12 i I 14 Fig. 4.62 Nondimensional vertical displacement profiles along the z-axis of a unit cell (a/b=l/2). 98 Chapter 5 CONCLUSIONS 5.1 Summary The major findings and conclusions of the present study are summarized in the following. 1) The displacement potential method together with a Bessel series solution of the relevant governing equations yields an exact analytical solution for fully coupled electroelastic field of a solid or annular piezoelectric cylinder under axisymmetric electromechanical loading. The Bessel series expansion of applied loading converges rapidly for smooth loading functions but shows very slow convergence and numerical instability near the discontinuities of discontinuous loading functions such as step functions. Overall, the solutions for electroelastic are found to be numerically stable and convergent when applied to solve a series of boundary-value problems involving solid and annular cylinders. Numerical solutions for stresses obtainedfromthe present study agree very closely with the existing solutions for the limiting case of an elastic cylinder. 2) Numerical results for solid and annular cylinders indicate that three-dimensional analysis is needed only if the length-radius ratio of a cylinder is less than two. In the case of long cylinders under vertical end loads or electric charge, vertical stress, vertical electric displacement and vertical electric field are uniform and very close to the one-dimensional solution. Both vertical displacement and electric field have their maximum values near the loading surface under applied vertical pressure. For a short cylinder the maximum value of vertical stress, electric displacement and electric field are more than three times the 1-D solution. 99 This has important implications in the design of short actuators. Vertical stress under electric charge loading has its maximum value near the loading end. 3) Electroelastic field in a solid or annular cylinder shows complex dependence on materials properties. For example, the magnitude of electric field generated in a PZT cylinder is substantially higher than that in a BaTi0 3 cylinder under both vertical and electric charge loading. On the other hand, vertical displacement is larger in the case of a BaTiO^ cylinder subjected to vertical loading. The two PZT materials show significantly different vertical stresses (one tensile the other compressive) and have higher displacements (stroke) under electric charge loading. The generation of tensile stresses in the actuator is a concern and has to be limited to the tensile strength of the ceramic. 4) The analytical formulation developed in this study for a unit cell of 1-3 piezocomposite provides an accurate representation of the electromechanical response under hydrostatic loading. Vertical stress in the piezoelectric phase in a unit cell could be much higher (four to eight times) than the applied pressure. This is a critical factor in the design of piezocomposites. Both vertical stress of the piezo phase and the interfacial shear stress increase with decreasing stiffness of the polymer matrix. 5) The present model is more accurate than the plane strain model and the approximate model of Li and Sottos [35]. Previous models do not account for the full electromechanical coupling in the piezo core. Effective properties are improved by using a softer matrix material but this has to be balanced against the increase in vertical stress of the piezo phase to avoid mechanical failure of the piezo core or interfacial failure. Based on the current results, PZT-5H shows better performance than PZT-4 for piezocomposite applications. For a given rod volume fraction, the performance of the composite can be increased by using a slender piezoelectric core. In addition, the best performance for effective 100 hydrostatic piezoelectric coefficient is obtained for a piezo phase volume fraction of 20-40%. The effective hydrostatic voltage coefficient has its maximum value for very low piezo volume fractions and could reach five to thirty times the value of a pure piezoelectric material for a relatively high aspect ratio, i.e. hja = 13.3. The optimum piezo phase volume fraction for a composite is around 15% based on the figure of merit. 5.2 Suggestions for Future Work To further understand the behavior of the class of problems considered in this thesis, the following recommendations are made for future work. 1) In the current solution, the shear stress boundary condition on the top and bottom surfaces of the unit cell is not satisfied. Based on a comparison with the finite element results, it is found that this approximation does not significantly affect the overall accuracy of numerical results. However, by exactly satisfying this condition a more accurate estimation of the interfacial shear stress can be obtained. 2) The solutions obtained for a unit cell can be extended to study the overall properties of a piezocomposite by applying relevant theories of micromechanics and homogenization techniques. 3) The general solutions for a cylinder derived in this study can be applied to study a range of problems involving fracture mechanics of solid and annular actuators and piezocomposites. At present, very little is known about the fracture mechanics of cylindrical actuators and piezocomposites. 4) Experimental studies are recommended to validate the theoretical study presented in this thesis and to enhance the overall understanding of the behaviour of piezoelectric actuators. Future studies should consider finite element modeling to 101 account for non-linear material behaviour and the role of electric fatigue actuators. 102 BIBLIOGRAPHY [I] Wada, B. K., Fanson, J. I. and Crawley, E. F., "Adaptive Structures" Mechanical Engineering, Vol. 112 (11), pp. 41 - 46, 1990. [2] Das, A. et al, "Adaptive structures: challenges, issues and opportunities", Proceedings of the 3u Conference on Decision and Control, pp. 2538-2542, 1991. [3] Chandra, J. et al, "Multidisciplinary research in smart structures: a survey", Proceedings of the American Control Conference, pp. 4167-4172, 1995. [4] Gandhi, M. V. and Thompson, B. S., Smart Materials and Structures, Chapman & Hall, London, 1992. [5] Crawley, E. F. and Luis, J., "Use of piezoelectric actuators as elements of intelligent structures", AIAA Journal, Vol. 25 (10), pp. 1373-1385, 1987. [6] Griffin, S. F. et al, "Piezoceramic sensors and actuators for smart composite structures", Proceedings of the 30 th Conference on Decision and Control, pp. 2543-2545, 1991. [7] Coulson, C. A. and Boyd, T. J. M., Electricity, Longman Inc., New York, 1979. [8] Dceda, T., Fundamentals of Piezoelectricity, Oxford University Press, New York, 1996. [9] Jaffe, B., Cook, W. R. and Jaffe, H., Piezoelectric Ceramics, Academic Press, New York, 1971. [10] Anderson, J. C , Dielectrics, Spottiswoode, Ballantyne & Co Ltd., London and Colchester, 1966. [II] Cheng, D. K., Field and Wave Electromagnetics 2 Addison-Wesley Pub. Co., 1989. [12] Crawley, E. F. and Lazarus, K. B., "Induced Strain Actuation of Isotropic and nd edn, Reading, Mass.: Anisotropic Plates" Proc. 30* AIAA/ASME/ASCEAHS/ASC Dynamics and Materials Conf, pp. 2000-2010, 1989. h [13] Structures, Structural Tamer, N. and Dahleh, M., "Feedback Control of Piezoelectric Tube Scanners", Proc. 33 Conf. of Decision and Control, pp. 1826- 1831, 1994. rd 103 [14] Cao, W., Zhang, Q . M . and Cross, L. E., "Theoretical study on the static performance of piezoelectric ceramic-polymer composites with 1-3 connectivity", J. Appl. Phys., Vol. 72 (12), pp. 5814-5821,1992. [15] Yu, N., "On overall properties of smart piezoelectric composites", Composites PartB: Engineering, Vol. 30, pp. 709-712, 1999. [16] Guinovart-Diaz, R. et al, "Overall properties of piezocomposite materials 1-3", Materials Letters, Vol. 48, pp. 93-98, 2001. [17] Levin, H. S. and Klosner, J. M., "Transverselly isotropic cylinders under band loads", Proc. ASCE: EM. J. Eng. Mech. Div., Vol. 93, pp. 157-174, 1967. [18] Atsumi, A. and Itou, S., "Stresses in a transversely isotropic circular cylinder having a spherical cavity", J. Appl. Mech., Vol. 41, pp. 507-511,1974. [19] Kasano, H. et al, "A transversely isotropic circular cylinder under concentrated loads", Bull. JSME, Vol. 23 (176), pp. 170-176,1980. [20] Kasano, H. et al, "Stresses and deformations in a transversely isotropic hollow cylinder under a ring of radial load", Bull, JSME, Vol. 25 (200), pp. 143-148, 1982. [21] Pickett, G., "Application of the fourier method to the solution of certain boundary problems in the theory of elasticity", J. Appl. Mech., Vol. 11, pp. 176-182, 1944. [22] Moghe, S. R. and Neff, H. F., "Elastic deformations of constrained cylinders", J. Appl. Mech., Vol. 38, pp. 393-399, 1971. [23] Power, L. D. and Childs, S. B., "Axisymmetric stresses and displacements in a finite circular bar", Int. J. Engng Sci., Vol. 9, pp. 241 -255, 1971. [24] Vendhan, C. P. and Archer, R. R., "Axisymmetric stresses in transversely isotropic finite cylinders", International Journal of Solids and Structures, Vol. 14, pp. 305- 318,1977. [25] Okumura, I. A., "Generalization of Elliott's solution to transversely isotropic solids and its application", Structural Eng. /Earthquake Eng., Vol. 4 (2), pp. 185195, 1987. [26] Okumura, I. A., "Stresses in a transversely isotropic, short hollow cylinder subjected to an outer band load", Ingenieur-Archiv, Vol. 59, pp. 310-324, 1989. [27] Wei, X. X., Chau, K. T. and Wong, R. H., "Analytic solution for axial point load strength test on solid circular cylinders", Journal of Engineering Mechanics, Vol. 125 (12), 1999. 104 [28] Chau, K. T., Wei, X. X., "Finite solid circular cylinders subjected to arbitrary surface load. Part I - Analytic solution", International Journal of Solids and Structures, Vol. 37, pp. 5707-5732, 2000. [29] Wei, X. X. and Chau, K. t., "Analytic solution for finite transversely isotropic circular cylinders under the axial point load test", Journal of Engineering Mechanics - ASCE, Vol. 128 (2), pp. 209 - 219, 2002. [30] Lekhnitskii, S. G., "Symmetrical deformation and torsion of a body of revolution with anisotropy of a special form", PMM-J. Appl. Math. Mech., Vol. 4 (3), pp. 4366, 1940. [31] Lekhnitskii, S. G., Theory of elasticity of an anisotropic elastic body, English translation by P. Fern, Holden-Day Inc., San Francisco, 1963. [32] Montgomery, R. E. and Richard, C , "A Model for the Hydrostatic Pressure Response of a 1-3 Composite", IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control, Vol. 43, pp. 456-466, 1996. [33] Wang, Z. and Zheng, B., "The General Solution of Three - Dimensional Problems in Piezoelectric Media", International Journal of Solids and Structures, Vol. 32, pp. 105 - 115, 1995. [34] Rajapakse, R. K. N. D. and Zhou, Y., "Stress Analysis of Piezoceramic Cylinders", Smart Materials and Structures, Vol. 6, pp. 169 - 177, 1997. [35] Li, L. and Sottos, N. R., "Improving hydrostatic performance piezocomposites", J. Appl. Phys., Vol. 77 (9), pp. 4595-4603, 1995. [36] Parton, V. Z. and Kudryavtsev, B. A., Electromagnetoelasticity, Gordon and Breach, New York, 1988. [37] Qi, H., Fang, D. and Yao, Z., "FEM analysis of electro-mechanical coupling effect of piezoelectric materials", Computational Materials Science, Vol. 8, pp. 283-290, 1997. [38] Bland, D. R., Solutions of Laplace's Equation, Routledge and Kegan Paul Ltd., London, 1961. [39] Watson, G. N., A Treatise on the Theory of Bessel Functions 2nd, Cambridge University Press, New York, 1995. [40] Newnham, R. E., Skinner, D. P. and Cross, L. E., "Connectivity and piezoelectricpyroelectric composites", Mat. Res. Bull., Vol. 13, pp. 525-536, 1978. 105 of 1-3 [41] Newnham, R. E., Bowen, L. J., Klicker, K. A. and Cross, L. E., "Composite piezoelectric transducer", Mater. Eng., Vol. 2 (12), pp. 93-106, 1980. [42] Gururaja, T. R., Newnham, R. E. and Cross, L. E., "Piezoelectric composite tranducers and sensors", in Electronic Ceramics, Lionel Levinson, Marcel Dekker Inc. New York, 1987. [43] Haun, M. J. and Newnham, R. E., "An experimental and theoretical study of 1-3 and 1-3-0 piezoelectric PZT-polymer composites for hydrophone applications", Ferroelectrics, Vol. 68, pp. 123-139, 1986. [44] Smith, W. A. and Shaulov, A. A., "Composite piezoelectrics: basic research to a practical device", Ferroelectrics, Vol. 87, pp. 309-320, 1988. [45] Smith, W. A., "The application of 1-3 piezocomposites in acoustic transducers", 1990 IEEE 7 International Symposium on Applications of Ferroelectrics, pp. 145-152, 1991. th [46] Ting, R. Y., "The hydroacoustic behavior of piezoelectric composite materials", Ferroelectrics, Vol. 102, pp. 215-224, 1990. • [47] Ting, R. Y., "A review on the development of piezoelectric composites for underwater acoustic transducer applications", IEEE Transactions on Instrumentation and Measurement, Vol. 41 (1), pp. 64-67, 1992. [48] Jensen, H., "Determination of macroscopic electro-mechanical characteristics of 1-3 piezoceramic/polymer composites by a concentric tube model", IEEE Trans. Ultrason. Ferroelec. Freq. Contr., Vol. 38 (6), pp. 591-594, 1991. [49] Schulgasser, K., "Relationships between the effective properties of transversely isotropic piezoelectric composites", J. Mech. Phys. Solids, Vol. 40 (2), pp. 473479, 1992. [50] Sottos, N. R., and Li, L., "Optimizing the Hydrostatic Properties of 1-3 Piezocomposites", Proc. SPIE - the International Society for Optical Engineering, Vol. 2189, pp. 50 - 61, 1995. [51] Li, L. and Sottos, N. R., "A design for optimizing the hydrostatic performance of 1-3 piezocomposites", Ferroelectric Letters, Vol. 21, pp. 41-46, 1996. [52] Little, R. W., Elasticity, Englewood Cliffs, N.J., Prentice-Hall, 1973. 106 APPENDIX A The constants Q. (i = 1,2,3,4) appearing in eqn (2.24) are defined by, Q =ec l5 x +£ c u u u •^2 ~&\\ \i "*"2e c +2f,,c + 2e c c 15 13 Q 3 = — £c 33 13 — 2e e c l3 l5 33 l5 33 — — \—2e e c 33 l5 £ c 3 3 n n ~£\\C c 33 n — 2f c — 2e +e c +2e c + e c — 2e c l3 l3 33 l3 13 33 33 n 15 33 15 33 "*"^33 33 11 ~*~^11 33 "*" 33 C C C i2 = —e — £ 4 33 3 3 C (Al) C33 The expressions for stresses and electric displacements in terms of yj are of the form, a ^ 2 "ar 2 1=1 V V 3 (C i=i VV a ^ 9 r 12 r 3r dz ia (A2) J N 2 + c,,-— 2 ^ n r dr (A3) dz, j 3 ~a^' 1=1 (A4) ,, drdz, "\2 3 (A5) where X, = (c, *» + * / ) M ' 3 2 A = {c k x3 +k u 2i ; v, = (c k $ = (1 + *,. + e * , ) / ^ ; g, = (e, + e, *„ - £ j , * 15 T, = [e k 33 u - £ k 33 2i 2 )H -1 5 5 33 2 / And / T (/ = 0,1,2,3) appearing in eqn (2.77) are in the form of 107 +e k 33 2i )/4 - c ; l3 ) / ' (i = 1,2,3) N u (A6) ft(^o-ft(a)^- r r _ 5 (C[i on = Zi pr )y ( sJo{Snri)+ _ )!/, Cl2 ( ) Cll Vl (A7) ^ , [ ( C „ - *3 )VW» fttefc r = A (V3 ) C r M«) ^VV* ( V 3 ) + ( . 2 " Cl 1 _ ( 7 ( ( H C 7 + n p C B 0 (c - c , , ) - £ , for,) n 12 (A8) (A9) ( n -Xi)y[^s I {s r,)+{c - c J - Z . f o r , ) c s im 0 = 1,2,3) 2 in eqns (2.115) and (2.116) are in the following form im in=fti n Q =- Vi r c OT 7i -C„)-/,(j /- )l + -/Jr2)V^2"^ o(V2) ( 12 4^i n ( i. + Xi\[^s K (s r,), (?, and R Ci L L s _ ,)I ( ) ) + ( H 0 n u (C„ -Z> ^o(C^)+(c,2 - C ) - ^ ( r r ) m n m t h +n ii 2 m fai -Xi)tJ»{t r)+{c -cj-^for) m 2 2 2 sin h(f h,)cos(nfr) m i ^-sinh{t hi) n (A10) m (All) r The constants JJ (j = 1,2,3) appearing in eqn (3.17) are in the form of /7,=l + c ; 77 =(l + c ) - c + l ; 2 12 2 2 1 2 77 =l + c 3 12 (A12) The expressions for stresses of the isotropic matrix in terms of (p. are of the form 2 ( ' ar n 2 i a •+ C V 12 2 2 N r ar • " a? 2 (A13) Pi _ a2 \ r 3r ^' dz i J i a 12 i=] S 3r 2 11 (A14) 2 (A15) drdz ; where 108 Xi l i; v = nPi c Mi = uPi c hi; =c p, Iv, - c ; 33 0 = 1,2) #={l+ )/^ 1 3 Pi (A16) Coefficients appearing in the system of linear algebraic equations of 1-3 piezocomposite are (A17) fe-zX&hfo)+(c: -d;Xte.b)\ (A18) 0"=u) 2 (A19) F _Px.n<Xm+7: ^3 =«. -«3 ^. ; n A n n =V4 [fe n «'4-=«2--a3»«'2»; J, KVVota, =V^fe -Z fc^T'ofe «.„ = —v V,. - fe + 0 ^ =£V.k«,) n )+fe-cf^ikoi)] )+('." Y ^K (C a, n (A20) n )] "^ (A21) 0=1,2,3) (A22) 0 = U) (A23) =1 23 )-fe - c.-^JC.fe )] (i = U) (A24) n-1 . n-l 1=1 0 >') ^ 1=1 109 2 2 n it V; (A25)
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Analysis of piezoelectric cylindrical actuators and...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Analysis of piezoelectric cylindrical actuators and 1-3 piezocomposite unit cells Chen, Yue 2003
pdf
Page Metadata
Item Metadata
Title | Analysis of piezoelectric cylindrical actuators and 1-3 piezocomposite unit cells |
Creator |
Chen, Yue |
Date Issued | 2003 |
Description | Adaptive structures incorporated with sensors and actuators are increasingly used in many engineering applications. Actuators used in adaptive structures are made out of piezoelectric ceramics, shape memory alloys (SMA), electrorheological fluids (ER fluids) or magnetostrictive materials. Among the many types of piezoelectric actuator elements, the cylindrical shape is widely used in practical applications involving fuel injectors, atomic force microscopes, high-precision telescopes, etc. In addition, the piezoelectric phase of piezocomposites is made out of cylindrical rods or fibers. Piezoelectric materials are very brittle and stress/electric field concentration at electrodes and other discontinuities often contribute to mechanical or dielectric breakdown. The study of electromechanical field of a cylindrical piezoelectric element and a composite unit cell with a piezoceramic core surrounded by a polymeric shell is therefore important to the understanding of failure of cylindrical actuators and design of piezocomposites for maximum electromechanical coupling. This thesis presents a comprehensive theoretical study of homogeneous piezoelectric cylinders and a unit cell of 1-3 piezocomposites. The governing equations for coupled axisymmetric electroelastic field in a transversely isotropic piezoelectric medium are established in terms of the displacements and electric potential. The general solutions of the governing equations are obtained in terms of a series of Bessel functions of the first and second kind. Several boundary-value problems are solved, and a computer code is developed to compute the electroelastic field in solid and annular cylinders for different aspect ratios, electromechanical loading and material properties. The salient features of the electroelastic field are identified. The effective properties of a 1-3 piezocomposite are studied under hydrostatic loading for different fiber volume fractions and polymer and ceramic properties. Optimum fiber volume fractions for maximum electromechanical coupling are determined for different ceramic-polymer combinations. |
Extent | 6298809 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-11-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0065643 |
URI | http://hdl.handle.net/2429/15350 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2004-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-ubc_2004-0113.pdf [ 6.01MB ]
- Metadata
- JSON: 831-1.0065643.json
- JSON-LD: 831-1.0065643-ld.json
- RDF/XML (Pretty): 831-1.0065643-rdf.xml
- RDF/JSON: 831-1.0065643-rdf.json
- Turtle: 831-1.0065643-turtle.txt
- N-Triples: 831-1.0065643-rdf-ntriples.txt
- Original Record: 831-1.0065643-source.json
- Full Text
- 831-1.0065643-fulltext.txt
- Citation
- 831-1.0065643.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0065643/manifest