Equations Of Linear Piezoelectricity Research Articles

A new two-dimensional theory for vibrations of piezoelectric crystal plates with electroded faces

A system of two-dimensional (2-D) governing equations for piezoelectric plates with general crystal symmetry and with electroded faces is deduced from the three-dimensional (3-D) equations of linear piezoelectricity by expansion in series of trigonometric functions of thickness coordinate. The essential difference of the present derivation from the earlier studies by trigonometrical series expansion is that the antisymmetric in-plane displacements induced by gradients of the bending deflection (the zero-order component of transverse displacement) are expressed by the linear functions of the thickness coordinate, and the rest of displacements are expanded in cosine series of the thickness coordinate. For the electric potential, a sine-series expansion is used for it is well suited for satisfying the electrical conditions at the faces covered with conductive electrodes. A system of approximate first-order equations is extracted from the infinite system of 2-D equations. Dispersion curves for thickness shear, flexure, and face-shear modes varying along x1 and those for thickness twist and face shear varying along x3 for AT-cut quartz plates are calculated from the present 2-D equations as well as from the 3-D equations, and comparison shows that the agreement is very close without introducing any corrections. Predicted frequency spectra by the present equations are shown to agree closely with the experimental data by Koga and Fukuyo [J. Inst. Elec. Comm. Engrs. of Japan 36, 59 (1953)] and those by Nakazawa, Horiuchi, and Ito [Proceedings of 1990 IEEE Ultrasonics Symposium (IEEE, New York, 1990)].

Governing equations for a piezoelectric plate with graded properties across the thickness

Two-dimensional first-order governing equations for electroded piezoelectric crystal plates with general symmetry and thickness-graded material properties are deduced from the three-dimensional equations of linear piezoelectricity by Mindlin's general procedure of series expansion. Mechanical displacements and thickness-graded material properties, i.e., the elastic stiffnesses, piezoelectric coefficients, dielectric permittivities, and mass density, are expanded in powers of the thickness coordinate, while electric potential is expanded in a special series in order to accommodate the specified electric potentials at electroded faces of the plate. The effects of graded material properties on the piezoelectrically induced stresses or deformations by the applied surface potentials are clearly exhibited in these newly derived equations which reduce to Mindlin's first-order equations of elastic anisotropic plates when the material properties are homogeneous. Closed form solutions are obtained from the three-dimensional equations of piezoelectricity and from the present two-dimensional equations for both homogeneous plates and bimorphs of piezoelectric ceramics. Dispersion curves for homogeneous plates and bimorphs and resonance frequencies for bimorph strips with finite width are computed from the solutions of three-dimensional and two-dimensional equations. Comparison of the results shows that predictions from the two-dimensional equations are very close to those from the three-dimensional equations.

A note on the static behavior of simply supported laminated piezoelectric cylinders

An exact three-dimensional solution of the equations of linear piezoelectricity is given for the static response of a finite laminated piezoelectric cylinder with its ends simply supported. The equilibrium equations and the conservation of charge equation are constructed in cylindrical coordinates. The three displacement components and the electrostatic potential are expressed in terms of Fourier series in the axial and circumferential directions, leading to a coupled system of four ordinary differential equations in the radial coordinate. The Frobenious method is used to obtain the elastic and electric fields for each layer of the laminate, which are a function of eight constants. Enforcing the boundary and continuity conditions across each interface yields the required number of equations to solve for these constants. Several examples are presented to study the fundamental behavior of these solids.

A laminated plate theory for high frequency, piezoelectric thin-film resonators

A high frequency, piezoelectric, laminated plate theory is developed and presented for the purpose of modeling and analyzing piezoelectric thin-film resonators and filters. The laminated plate equations are extensions of anisotropic composite plate theories to include piezoelectric effects and capabilities for modeling harmonic overtones of thickness-shear vibrations. Two-dimensional equations of motion for piezoelectric laminates were deduced from the three-dimensional equations of linear piezoelectricity by expanding the mechanical displacements and electric potential in a series of trigonometric function, and obtaining stress resultants by integrating through the plate thickness. Relations for handling the mechanical and electrical effects of platings on the top and bottom surfaces of the laminate are derived. A new matrix method of correcting the cutoff frequencies is presented. This matrix method could also be used to efficiently correct the cutoff frequencies of any nth order plate laminate theories which employ Mindlin’s form of polynomial expansion of mechanical displacements and electric potential through the plate thickness. The first order laminated plate theory with correction factors for cutoff frequencies and slope of the flexural branch at large frequencies was applied to a 2-layer, zinc oxide-silicon strip, and a 3-layer, zinc oxide-zinc oxide-silicon strip. Open circuit, dispersion relations were generated for a range of volume fractions and compared to the exact dispersion relations. Both the 2-layer and 3-layer strip show similar qualitative comparison: The present theory compares fairly well with the exact dispersion relation for real wave numbers and nonpropagating (imaginary) wave numbers which are smaller than 0.5i. The extensional branch, and thickness-shear branches begin to deviate from the exact solution when the nondimensionalized frequency is greater than one. Consequently, to maintain accuracy of solutions when using the present first order laminated plate theory, one should limit the calculation of resonant, nondimensionalized frequencies to less than 1.1. Results for the frequency spectrum in the vicinity of the open circuit, fundamental thickness-shear frequency, and modes shapes were presented for the 2-layer strip with fixed edges, and a volume fraction of silicon equal to 0.2. The technically important fundamental thickness-shear mode is found to have the shear component strongly coupled with extensional component.

Electromagnetic radiation from an AT-cut quartz plate under lateral-field excitation

Equations of linear piezoelectricity (with the quasistatic approximation) for the quartz plate and Maxwell’s equations for the electromagnetic field in the surrounding vacuum are solved for the thickness-shear vibrations of rotated Y cut of quartz plate. For an AT-cut plate vibrating near the thickness-shear resonance excited by a uniform, lateral electric field of magnitude 1 V/m, the electromagnetic energy radiated from each face is about 0.13 μW/cm2. The radiated power is about 0.1 μW/cm2 if the plate is excited by a shearing face traction which produces a strain of 10−5. Present solution is compared in detail with Mindlin’s [Int. J. Solids Struct. 9, 697 (1972)] solution of equations of piezoelectromagnetism (without the quasistatic approximation) for the thickness-shear vibrations excited by shearing face traction. It is found that the percent difference in radiated powers computed from these two solutions, due to the quasistatic approximation, is in the order of β2(=v̂2/ĉ2), where v̂ is the velocity of the x1-thickness-shear wave, ĉ is the velocity of electromagnetic wave of E3 propagating in the x2 direction in quartz, and β≊10−5.

A two-dimensional theory for high-frequency vibrations of piezoelectric crystal plates with or without electrodes

Two-dimensional equations of motion of successively higher-order approximations for piezoelectric crystal plates with triclinic symmetry are deduced from the three-dimensional equations of linear piezoelectricity by expansion in series of trigonometric functions of the thickness coordinate of the plate. These equations, complemented by two additional relations: one, the usual relation of face tractions to the mass of electrodes, and the other relating face charges to face potentials and face displacements, can accommodate either the traction and charge boundary conditions at the faces of the plate without electrodes or the traction and potential boundary conditions at the faces of the plate with electrodes. Dispersion curves are obtained from the first- to fourth-order approximate plate equations for a rotated 45° Y-cut lithium tantalate plate without electrodes, and these curves are compared with those from the frequency equation of the three-dimensional equations with close agreement. Solutions of forced vibrations of an AT-cut quartz plate with electrodes are obtained from the first-order plate equations complemented by the two additional relations. It is shown that cutoff frequencies of the fundamental thickness-shear vibrations from the approximate equations are identical to the ‘‘exact’’ ones obtained from the three-dimensional equations.

An analysis of doubly rotated quartz resonators utilizing essentially thickness modes with transverse variation

Closed-form asymptotic expressions for the frequency–wavenumber dispersion relations in doubly rotated quartz plates vibrating in the vicinity of the odd pure thickness frequencies are derived from the equations of linear piezoelectricity and the associated boundary conditions on the major surfaces. The usual assumptions of small piezoelectric coupling and small wavenumbers along the plate are made and it is supposed that the pure thickness frequencies are sufficiently different that one pure thickness wave is dominant at a time. In the treatment the mechanical displacement is decomposed along the eigenvector triad of the pure thickness solution to facilitate the asymptotic analysis. The fact that the wavenumbers along the plate are restricted to be small significantly reduces the complexity of the equations without neglecting any transformed elastic constants. The resulting asymptotic dispersion equation enables the construction of a scalar differential equation describing the transverse behavior of essentially thickness modes of vibration in doubly rotated quartz plates. The scalar equation is applied in the analysis of both trapped energy resonators with rectangular electrodes and contoured crystal resonators using established procedures. In particular, calculations performed for the contoured SC cut and a number of other doubly rotated orientations are shown to be in excellent agreement with experiment. Since the differential equation for each harmonic family depends on the order of the harmonic and in the general doubly rotated case contains mixed derivatives in the plane of the plate, a different transformation is required for each harmonic family to obtain the coordinate system in which the mixed derivatives do not appear and, hence, the equation is separable. An interesting consequence of this transformation is that since the nodal planes of the anharmonics of each harmonic family of the contoured SC-cut quartz resonator are oriented along the transformed coordinate system for that harmonic family, they are oriented differently for each harmonic family.

Investigation of the amplitude distribution of AT-cut quartz crystals

The three-dimensional equations of linear piezoelectricity are used to compute the mechanical vibration amplitude distributions, as well as the resonance frequencies of plano-convex circular AT-cut quartz crystal resonators. These computations are compared with vibration amplitude measurements employing a new technique that utilizes the speckle effect. Very good agreement has been found between the measured and the computed vibration amplitude distributions.

Frequencies of piezoelectrically forced vibrations of electroded, doubly rotated, quartz plates

Two-dimensional equations of motion or piezoelectric crystal plates, obtained from the three dimensional equations of linear piezoelectricity by expansion in power series of the thickness coordinate of the plate, are solved for forced vibrations of electroded SC-cut quartz plates. Results of computations are given for frequencies of simple thickness modes of vibration, for the dispersion of straight-crested waves and for frequencies of vibration of a strip, along with its dimensional ratios for minimal coupling of the fundamental thickness-shear mode with overtones of flexure, face-shear and thickness-twist.

PIEZOELECTRIC TRANSDUCER DISC RESONANCES

Abstract The electrical impedance characteristics of a piezoelectric disc are related to the mechanical resonance phenomena in the disc. The theoretical electrical impedance evaluated from the linear piezoelectric equations will only predict resonant properties for certain discs. In other discs additional ‘resonances’ occur due to the excitation of extra modes. Such resonances have been experimentally measured and theoretically investigated.

Electroelastic interactions and the piezoelectric equations

The relation between the equations of linear piezoelectricity and the more general nonlinear electroelastic equations is discussed. The influence of nonlinear deformation theory and rotational invariance is noted. The linear equations for small dynamic fields superposed on a static bias are presented. An equation for the perturbation of the eigenvalue of the piezoelectric solution due to a bias is exhibited. Some applications of the perturbation equation are given.

Electrostatic interactions and the piezoelectric equations

In piezoelectricity, the stress equations of motion are coupled to the charge equation of electrostatics by means of the linear piezoelectric constitutive equations which contain the piezoelectric constants that couple the small strain to the electric field. In the case of the more general nonlinear interaction of the electric field with a deformable insulator, the Maxwell electrostatic stress tensor must be included in the description. Furthermore, in the nonlinear case, a rotationally invariant combination of the electric field and the deformation occurs in the constitutive equations in place of the electric field for the same reason that the finite train occurs. The linear electroelastic equations for small dynamic fields, superposed on a static bias obtained from the nonlinear system, are more general than the linear piezoelectric equations in that the linear constitutive equations depend on the small local mechanical rotation, as well as the small strain. Nevertheless, when the biasing stress and ele...

Perturbation theory for linear electroelastic equations for small fields superposed on a bias

A perturbation formulation of the equations of linear piezoelectricity for small fields superposed on a bias is obtained from a Green’s function representation. It is shown that the resulting equation for the first perturbation of the eigenvalue may be obtained without the use of a Green’s tensor or a complete set of orthogonal eigensolutions. Since the bias enters the constitutive equations, the boundary conditions contain perturbation terms as well as the differential equations. The linear electroelastic equations for small fields superposed on a bias differ from the equations of linear piezoelectricity because the effective material constants of linear electroelasticity have less symmetry than the constants of linear piezoelectricity. Consequently, a perturbation formulation of the linear electroelastic equations for small fields superposed on a bias is presented. It is shown that the effective constants of linear electroelasticity have just the symmetry required for the condition of the orthogonality of linear electroelastic vibrations to hold.

A perturbation analysis of the attenuation and dispersion of surface waves

A perturbation formulation of the equations of linear piezoelectricity is obtained using a Green’s function approach. Although the resulting equation for the first perturbation of the eigenvalue strictly holds for real perturbations of real eigenvalues only, it is formally extended to the case of purely imaginary perturbations of real eigenvalues. The extended equation is applied in the calculation of the attenuation of surface waves due to the finite electrical conductivity of thin metal films plated on the surface and air loading. The influence of the viscosity of the air is included in the air-loading analysis, and the calculated attenuation increases accordingly. Since the metal films are thin compared with a wavelength, an approximate thin-plate conductivity equation is employed in the determination of the attenuation due to the electrical conductivity of the films. The resulting attenuation is obtained over a very large range of values of sheet conductivity. This is accomplished by using the equation for the first perturbation of the eigenvalue iteratively to determine the solution and attendant attenuation to any desired degree of accuracy. The phase velocity dispersion curve due to the mechanical effect of a thin film plated on a substrate is determined for relatively large wavelengths by employing the perturbation equation iteratively, and excellent agreement is obtained with the results of other more direct approaches. The calculations have been performed for an aluminum film on either ST-cut quartz or Y-Z lithium niobate.

Nonlinear electroelastic equations for small fields superposed on a bias

The nonlinear differential equations and boundary conditions in small-field variables, for small fields superposed on large static biasing states, are obtained from general rotationally invariant nonlinear electroelastic equations derived previously. The small-field equations are directly applicable in the consistent description of parametric effects in high-coupling piezoelectric materials in terms of the fundamental material parameters. The application of the equations to homogeneously polarized ferroelectrics reveals that in the linear limit the electroelastic equations are identical with the equations of linear piezoelectricity for the symmetry of the polarized state. The influence of a thickness directed homogeneous biasing electric field on the thickness vibrations of a piezoelectric plate, to second order in the biasing field, has been determined. To first order in the biasing field the results indicate that the effective fifth-rank tensor assumed in earlier quasilinear work on the subject did not have correct symmetry properties because the influence of the homogeneous static deformation under the biasing field was ignored.

High-Frequency Vibrations of Crystal Surfaces, Plated with Electrodes

In a recent paper (M. C. Dökmeci, to be published in J. Math. Phys.), we established a variational theorem for linear piezoelectricity and then constructed a theory of piezoelectric crystal surfaces. This theory is now extended to include the effects of the elastic stiffness and inertia of the electrodes in crystal shells completely coated with electrodes on both faces. Each of the electrodes may have different thickness and material properties, and each is assumed to be perfectly conducting. Starting with the local equations of linear piezoelectricity and using an averaging procedure, we establish the two-dimensional dynamical theory of plated crystal surfaces for the case when the mechanical displacement and electric potential fields are taken to vary linearly with respect to the thickness coordinate. The interactions between the crystal shell and its electrodes are taken into account. Further, we study some special cases of practical importance such as very thin electrodes and crystal plates [cf. R. D. Mindlin, in Progress in Applied Mechanics (Macmillan, New York, 1963), p. 73, and H. F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969)], and present the theorems of uniqueness in this theory. [Work supported by the U. S. Office of Naval Research.]

Electromechanical coupling factors and fundamental material constants of thickness vibrating piezoelectric plates

The vibrations of piezoelectric plates are governed by the linear piezoelectric equations which may contain a rather large number of elastic, piezoelectric and dielectric constants. But under certain very restrictive but practically important circumstances, piezoelectric coupling coefficients occur naturally in the solutions of the equations, a different coefficient arising in each particular case. Under more general circumstances many of the material constants remain in the solution, and the utility or significance of any of the coupling coefficients appears questionable. The solution of the problem of the thickness vibrations of an arbitrarily anisotropic plate is used in demonstrating the significance and limitations of the coupling coefficient for various types of anisotropy. In many of the instances when the coupling coefficient turns out to be useful, the frequency equation shows that the overtone resonances are not integral multiples of the fundamental. The departure from an integral multiple relationship depends on the coupling coefficient only, and consequently, the coupling coefficient can be determined from simple resonance measurements.

Flexural Vibrations of Bilaminar Ceramic Plates

The differential equation and boundary conditions that govern the electrically forced flexural vibrations bilaminar ceramic plates are developed from linear piezoelectric equations of state. A general formula for the electrical-input admittance of a bilaminar plate is given. The differential equation can be cast in the classical form for thin plates by defining a “modified flexural rigidity” and a “modified ratio.” The applied voltage does not appear in the differential equation as a forcing term; instead, it enters the problem through the boundary conditions as a uniform applied bending moment at any nonclamped edge of the plate. The effects electromechanical coupling are illustrated by solving the equation in the case of a free-edge disk. The resonance frequencies (for constant applied current) and the vibration patterns at these frequencies vary with electromechanical coupling; hence, the coupling coefficient of the plate material can be inferred from simple careful measurements of resonance frequencies. The effects of rotary inertia and shear on the properties of these plates are indicated. [Work supported in part by U. S. Office of Naval Research.]

Wave Propagation in an Infinite Piezoelectric Plate

The problem of wave propagation in an infinite piezoelectric plate belonging to crystallographic class C6v, situated between shorted electrodes, is rigorously analyzed for two special orientations of the sixfold axis. The solution is derived using the linear piezoelectric equations. The analysis shows that for a given frequency and wave number in the propagation direction there are three independent solutions of the differential equations, and, furthermore, that these three solutions couple at the traction-free boundaries of the plate. The dispersion curves can be calculated from the resulting transcendental equations. It has previously been shown that the resonant frequencies at infinite wavelength of a plate, in which the electromechanical coupling factors are high, deviate considerably from those of the purely elastic solution. Thus the dispersion spectrum calculated from this analysis will deviate considerably from that of the purely elastic analysis. It is shown in an appendix that the solution for the plate of infinite width is also applicable in the case of the plate of infinitesimal width.

Thickness Vibrations of Piezoelectric Plates

The thickness vibrations of an infinite anisotropic plate with electrodes coated on both surfaces are investigated using the linear piezoelectric equations. In general, the three fundamental solutions of the differential equations are found to couple at the traction-free surfaces of the plate. The solution reduces to the well-known uncoupled solution for the purely elastic case when the piezoelectric constants are zero. The transcendental equation which determines the resonant frequencies is derived. The theory is applied to a ferroelectric ceramic in two special cases; one with the plate polarized in the thickness direction, and the other with the plate polarized in its plane. The theory is also applied to a rotated Y-cut quartz plate. In all of these special cases the three solutions of the differential equations uncouple. Two of these solutions are identical with the purely elastic solutions and cannot be excited electrically, whereas the third differs from the purely elastic solution and can be excited electrically. The frequency equation obtained from this piezoelectric solution shows that the resonant frequencies are not integral multiples of the fundamental.