DYNAMIC ANTI-PLANE PROBLEMS OF PIEZOCERAMICS AND APPLICATIONS IN ULTRASONICS—A REVIEW

We review theoretical results on anti-plane motions of polarized ceramics based on the linear theory of piezoelectricity. Solutions to dynamic problems of the propagation of bulk acoustic waves （BAW） and surface acoustic waves （SAW）, vibrations of finite bodies, and applications to various piezoelectric devices including piezoelectric waveguides, resonators, mass sensors, fluid sensors, actuators, nondestructive evaluation, power harvesters （generators）, transformers, and power transmission through an elastic wall by acoustic waves are discussed. Complications due to material inhomogeneity, initial stress, electromagnetic coupling, electric field gradient and semiconduction are also discussed. The paper cites 82 references.

Analysis of thickness-shear and thickness-twist modes of AT-cut quartz acoustic wave resonator and filter

The thickness-shear （TS） and thickness-twist （TT） vibrations of partially electroded AT-cut quartz plates for acoustic wave resonator and filter applications are theoretically studied. The plates have structural variations in one of the two in-plane directions of the plates only. The scalar differential equations derived by Tiersten and Smythe for electroded and unelectroded AT-cut quartz plates are used, resulting in free vibration resonant frequencies and mode shapes for both fundamental and overtone fam- ilies of modes. The trapped modes with vibrations, mainly confined in the electroded areas, are found to exist in both the resonator and the filter structures. The numerical results for the trapped modes are presented for different aspect ratios of electrodes and material properties, providing a reference to the design and optimization of quartz acous- tic wave resonators and filters.

Stress-induced potential barriers and charge distributions in a piezoelectric semiconductor nanofiber

The performance of a piecewise-stressed ZnO piezoelectric semiconductor nano?ber is studied with the multi-?eld coupling theory. The ?elds produced by equal and opposite forces as well as sinusoidally distributed forces are examined. Speci?c distributions of potential barriers, wells, and regions with effective polarization charges are found. The results are fundamental for the mechanical tuning on piezoelectric semiconductor devices and piezotronics.

Two-dimensional equations for thin-films of ionic conductors

A theoretical model is developed for predicting both conduction and diffusion in thin-film ionic conductors or cables. With the linearized Poisson-Nernst-Planck(PNP)theory, the two-dimensional(2D) equations for thin ionic conductor films are obtained from the three-dimensional(3D) equations by power series expansions in the film thickness coordinate, retaining the lower-order equations. The thin-film equations for ionic conductors are combined with similar equations for one thin dielectric film to derive the 2D equations of thin sandwich films composed of a dielectric layer and two ionic conductor layers. A sandwich film in the literature, as an ionic cable, is analyzed as an example of the equations obtained in this paper. The numerical results show the effect of diffusion in addition to the conduction treated in the literature. The obtained theoretical model including both conduction and diffusion phenomena can be used to investigate the performance of ionic-conductor devices with any frequency.

HIGH-FREQUENCY VIBRATIONS OF CORRUGATED CYLINDRICAL PIEZOELECTRIC SHELLS

Coupled extensional and flexural cylindrical vibrations of a corrugated cylindrical piezoelectric shell consisting of multiple pieces of circular cylindrical surfaces smoothly connected along their generatrix are studied. To validate the results for the case of relatively thick shells or equivalently high-frequency modes with short wavelengths, existing analysis is extended by considering shear deformation and rotatory inertia. An analytical solution is obtained. Based on the solution, resonant frequencies and mode shapes are calculated.

MECHANICAL BEHAVIOUR OF NATURAL COW LEATHER IN TENSION

We study experimentally the microstructure and mechanical behavior of natural cow leather. Tensile tests are performed using leather strips to observe their deformation, creep and failure. It is found that the microstructure of cow leather is a layered, complicated network of fibers of different sizes from a few to a few hundreds of nanometers in diameter. They show nonlinear stress-strain relations and viscoelastic behavior. The effect of humidity is also examined. A simple theoretical model of a multilayered beam is proposed to describe the most basic behavior in tension.

A semi-analytical solution for electric double layers near an elliptical cylinder

A theoretical analysis on the electric double layer formed near the surface of an infinite cylinder with an elliptical cross section and a prescribed electric potential in an ionic conductor was performed using the linearized Gouy–Chapman theory. A semi-analytical solution in terms of the Mathieu functions was obtained. The distributions of the electric potential, cations, anions, and electric field were calculated. The effects of various physical and geometric parameters were examined. The fields vary rapidly near the elliptical boundary and are nearly uniform at far field. Electric field concentrations were found at the ends of the semi-major and semi-minor axes of the ellipse. These concentrations are sensitive to the physical and geometric parameters.

ELECTRICALLY FORCED THICKNESS-SHEAR VIBRATIONS OF QUARTZ PLATE WITH NONLINEAR COUPLING TO EXTENSION

We study electrically forced nonlinear thickness-shear vibrations of a quartz plate resonator with relatively large amplitude. It is shown that thickness-shear is nonlinearly coupled to extension due to the well-known Poynting effect in nonlinear elasticity. This coupling is relatively strong when the resonant frequency of the extensional mode is about twice the resonant frequency of the thickness-shear mode. This happens when the plate length/thickness ratio assumes certain values. With this nonlinear coupling, the thickness-shear motion is no longer sinusoidal. Coupling to extension also affects energy trapping which is related to device mounting. When damping is 0.01, nonlinear coupling causes a frequency shift of the order of 10^-6 which is not insignificant,and an amplitude change of the order of 10^-8. The effects are expected to be stronger under real damping of 10^-5 or larger. To avoid nonlinear coupling to extension, certain values of the aspect ratio of the plate should be avoided.

Effects of surface impedance on current density in a piezoelectric resonator for impedance distribution sensing

We study the relationship between the surface mechanical load represented by distributed acoustic impedance and the current density distribution in a shear mode piezoelectric plate acoustic wave resonator.A theoretical analysis based on the theory of piezoelectricity and trigonometric series is performed.In the specific and basic case when the surface load is due to a local mass layer,numerical results show that the current density concentrates under the mass layer and is sensitive to the physical as well as geometric parameters of the mass layer such as its location and size.This provides the theoretical foundation for predicting the surface impedance pattern from the current density distribution,which is fundamental to the relevant acoustic wave sensors.

Electromechanical Fields Near a Circular PN Junction Between Two Piezoelectric Semiconductors

We study electromechanical fields near the interface between a circular piezoelectric semiconductor cylinder and another piezoelectric semiconductor in which it is embedded. The cylinder is p-doped. The surrounding material is n-doped. The phenomenological theory of piezoelectric semiconductors consisting of the equations of piezoelectricity and the conservation of charge for holes and electrons is used. The theory is linearized for small carrier concentration perturbations. An analytical solution is obtained, showing the formation of a PN junction near the interface. Various electromechanical fields associated with the junction are calculated. The effects of a few physical parameters are examined.

Transient Bending Vibration of a Piezoelec trie Semiconductor Nanofiber Under a Suddenly Applied Shear Force

We study transient bending vibration of a ZnO piezoelec trie semiconduc tor nanofiber fixed at one end with a suddenly applied shear force at the other end.A one-dimensional model based on the phenomenological t heory of piezoelec trie semiconductors consisting of the equations of piezoelectricity coupled to the continuity equation of electrons is used.An approximate theoretical analysis is performed,accompanied by a finite element analysis using COMSOL.The evolutions of deflection,elec trie pot ential and elec tron dist ribution are calculated and examined.It is found that when the fiber reaches its largest deflection,the distributions of the electromechanical fields are qualitatively similar to those in the case of static loading under the same shear force,but the amplitudes of the fields are about twice as large roughly.

Electrical Response of a Multiferroic Composite Semiconductor Fiber Under a Local Magnetic Field

We study the electrical response of a multiferroic composite semiconductor fiber consisting of a piezoelectric semiconductor layer and two piezomagnetic layers under a transverse magnetic field applied locally to a finite part of the fiber.The phenomenological theory of piezomagnetic-piezoelectric semiconductors is employed.A one-dimensional model is derived for magnetically induced extension of the fiber.For open-circuit boundary conditions at the two ends of the fiber,an analytical solution is obtained from the model linearized for small carrier perturbations.The solution shows a local electric polarization and a pair of local electric potential barrier-well.When the two ends of the fiber are under a voltage,a nonlinear numerical solution shows that the potential barrier and well forbid the passage of currents when the voltage is low.The results have potential applications in piezotronic devices when magnetic fields are involved for manipulating the devices or sensing and transduction.

EFFECTS OF NONLINEARITY ON TRANSIENT PROCESSES IN AT-CUT QUARTZ THICKNESS-SHEAR RESONATORS

We study the effects of mechanical nonlinearity arising from large thickness-shear deformation on the transient process of an AT-cut quartz plate resonator. Mindlin＇s two-dimensional plate equation is used, from which a system of first-order nonlinear differential equations governing the evolution of the vibration amplitude is obtained. Numerical solutions by the Runge-Kutta method show that in common operating conditions of quartz resonators the nonlinear effect varies from noticeable to significant. As resonators are to be made smaller and thinner in the future with about the same power requirement, nonlinear effects will become more important and need more understanding and consideration in resonator design.

Bending of a Flexoelectric Semiconductor Plate

We study electromechanical responses of a flexoelectric semiconductor plate in bending under mechanical loads.A two-dimensional theory for classical bending without shear deformation is derived from the three-dimensional macroscopic theory of flexoelectric semiconductors.A simple solution is obtained for pure bending.A combination of physical and geometric parameters is introduced as a measure of the strength of the coupling between the mechanical load and the redistribution of charge carriers.A trigonometric series solution is obtained for a simply supported rectangular plate under a local normal mechanical load,which shows concentration of mobile charges and the formation of electric potential barriers near the loading area.The results are fundamental to the mechanical manipulation of charge carriers in these plates.We also analyze the buckling of a simply supported rectangular plate under in-plane compressive forces.A series of buckling loads and modes are obtained.Numerical results show that flexoelectric coupling exhibits a stiffening effect and increases the buckling load,while semiconduction weakens the flexoelectric stiffening.The distributions of mobile charges in the first few buckling modes are presented.

THICKNESS-SHEAR AND THICKNESS-TWIST VIBRATIONS OF CIRCULAR AT-CUT QUARTZ RESONATORS

Exact solutions for free vibration frequencies and modes are obtained for thickness- shear and thickness-twist vibrations of unelectroded circular AT-cut quartz plates governed by the two-dimensional scalar differential equation derived by Tiersten and Smythe. Comparisons are made with experimental results and the widely-used perturbation solution by Tiersten and Smythe under the assumption of weak in-plane anisotropy. Our solution is found to be much closer to the experimental results than the perturbation solution. For the frequency of the fundamental thickness-shear mode, the error of the perturbation method is 0.4549%, significant in resonator applications.

THICKNESS-SHEAR VIBRATION OF A RECTANGULAR QUARTZ PLATE WITH PARTIAL ELECTRODES

We study free vibration of a thickness-shear mode crystal resonator of AT-cut quartz. The resonator is a rectangular plate partially and symmetrically electroded at the center with rectangular electrodes. A single-mode, three-dimensional equation governing the thickness-shear displacement is used. A Fourier series solution is obtained. Numerical results calculated from the series show that there exist trapped thickness-shear modes whose vibration is mainly under the electrodes and decays rapidly outside the electrodes. The effects of the electrode size and thickness on the trapped modes are examined.

ENERGY TRAPPING OF THICKNESS-SHEAR AND THICKNESS-TWIST MODES IN A PARTIALLY ELECTRODED AT-CUT QUARTZ RESONATOR

The thickness-shear and thickness-twist vibrations of a finite and partially elec- troded AT-cut quartz resonator are investigated. The equations of anisotropic elasticity are used with the omission of the small elastic constant c56. An analytical solution is obtained using Fourier series from which the free vibration resonant frequencies, mode shapes, and energy trapping are calculated and examined.

An Analysis of Piezomagnetic-Piezoelectric Semiconductor Unimorphs in Coupled Bending and Extension under a Transverse Magnetic Field

We study the response of a composite beam(unimorph)with a piezomagnetic layer and a piezoelectric semiconductor layer under a transverse magnetic field.A one-dimensional model for coupled bending and extension of the beam is derived from the three-dimensional theory.A simple and analytical solution of the model is obtained,showing that various electromechanical fields develop in the beam through piezomagnetic and piezoelectric couplings as well as semiconduction.In particular,the mobile charges in the semiconductor layer redistribute themselves under the magnetic field.A coupling coefficient is defined to describe the strength of this magneto-semiconduction coupling.The effects of physical and geometric parameters on the fields and the coupling coefficient are examined.The results are fundamental to piezotronics when magnetic effects are involved.

Differential derivation of momentum and energy equations in electroelasticity

This paper presents a derivation of the equations of linear momentum, angular momentum, and energy of an electroelastic body using a composite particle consisting of two differential elements based on Tiersten＇s two-continuum model. The differential derivation shows the physics involved in a way different from the integral approach in the literature. Like the integral approach, it also produces the expressions of the electric body force, couple, and power which are fundamental to the development of the nonlinear macroscopic theory of an electroelastic body.

Mechanical Manipulation of Electrical Behaviors of Piezoelectric Semiconductor Nanofibers by Time-Dependent Stresses

We study electric currents in a piezoelectric semiconductor fiber under a constant voltage and time-dependent axial stresses applied locally.From a nonlinear numerical analysis based on a one-dimensional phenomenological model using the commercial software COMSOL,it is found that pulse electric currents can be produced by periodic or time-harmonic stresses.The pulse currents can be tuned by the amplitude and frequency of the applied stress.The result obtained provides a new approach for the mechanical control of electric currents in piezoelectric semiconductor fibers and has potential applications in piezotronics.