A variational principle for the equations of viscopiezoelectricity.

The three-dimensional equations of linear viscopiezoelectricity and an accompanying electromechanical energy theorem are deduced, by the quasielectrostatic approximation, from the equations of viscoelectromagnetism and a generalized Poynting's theorem, respectively. For a viscopiezoelectric solid of volume V and bounding surface S, the internal energy, kinetic energy, and electric enthalpy densities as well as the variation of work done over S and the variation of energy dissipation in V are defined. A variational principle in terms of the defined functions is presented. It is shown that, from the principle, the equations of viscopiezoelectricity in V and the natural boundary conditions on S are obtained.

Effects of a liquid layer on thickness-shear vibrations of rectangular AT-cut quartz plates.

Thickness-shear vibrations of a rectangular AT-cut quartz with one face in contact with a layer of Newtonian (linearly viscous and compressible) fluid are studied. The two-dimensional (2-D) governing equations for vibrations of piezoelectric crystal plates given previously are used in the present study. The solutions for 1-D shear wave and compressional wave in a liquid layer are obtained, and the stresses at the bottom of the liquid layer are used as approximations to the stresses exerting on the crystal surface in the plate equations. Closed form solutions are obtained for both free and piezoelectrically forced thickness-shear vibrations of a finite, rectangular AT-cut quartz in contact with a liquid layer of finite thickness. From the present solutions, a simple and explicit formula is deduced for the resonance frequency of the fundamental thickness-shear mode, which includes the effects of both shear and compressional waves in the liquid layer and the effect of the thickness-to-length ratio of the crystal plate. The formula reduces to the widely used frequency equation obtained by many previous investigators for infinite plates. The resonance frequency of a rectangular AT-cut quartz, computed as a function of the thickness of the adjacent liquid layer, agrees closely with the experimental data measured by Schneider and Martin.

Mechanical effects of electrodes on the vibrations of quartz crystal plates.

A system of approximate first-order equations is extracted from an infinite system of 2-D equations for piezoelectric crystal plates with thickness-graded material properties, which is deduced from the 3-D equations of linear piezoelectricity. These equations are used to study mechanical effects on the thickness-shear (TS), flexural (F), and face-shear (FS) vibrations of an AT-cut quartz plated with two identical electrodes. Dispersion curves are calculated from the present 2-D equations as well as the 3-D equations. The comparison of these curves shows that the agreement is very close for all three frequency branches of TS, F, and FS modes in a range up to the 1.5 times the fundamental TS frequency and for gold and aluminum electrodes with R, the ratio of the mass of the electrodes to that of the plate, equal to 0.05. without introducing any correction factors. In order to assess electrode effects, spectra of omega vs. a/bq (length-to-thickness ratio of the quartz) are computed for plates with gold and aluminum electrodes and different R ratios. And the spectrum of omega vs. R is computed for plates with aluminum electrodes and a given a/bq ratio. For a plate with gold electrodes, the frequencies of predominant TS, F, and FS modes are decreasing as R increases, but the amount of frequency changes for the TS mode is much greater than those for the other two modes. However, for a plate with aluminum electrodes, the frequencies of the TS and FS modes are decreasing, but those of the F modes are increasing as R increases.

Plane harmonic waves in an infinite piezoelectric plate with dissipation.

In a previous paper, the three-dimensional equations of linear piezoelectricity with quasielectrostatic approximation were extended to include losses attributed to the mechanical damping in solid and the resistance in current conduction. These equations were used to investigate the plane wave propagation in an unbounded solid and forced thickness vibration of an infinite piezoelectric plate. In the present paper, these equations are used to obtain solutions of plane harmonic wave of arbitrary direction in an infinite and dissipative piezoelectric plate with general crystal symmetry. Dispersion curves are computed and plotted for real frequencies and complex wave numbers. All frequency branches are complex for dissipative plate. There are no longer any pure real or pure imaginary or complex conjugate frequency branches as those existing for nondissipative plates. Effects of dissipation on the wave propagation are examined in detail for AT-cut of quartz as well as barium titanate ceramic plate.

Thickness vibrations of a piezoelectric plate with dissipation.

The three-dimensional (3-D) equations of linear piezoelectricity with quasi-electrostatic approximation are extended to include losses attributed to the acoustic viscosity and electrical conductivity. These equations are used to investigate effects of dissipation on the propagation of plane waves in an infinite solid and forced thickness vibrations in an infinite piezoelectric plate with general symmetry. For a harmonic plane wave propagating in an arbitrary direction in an unbounded solid, the complex eigenvalue problem is solved from which the effective elastic stiffness, viscosity, and conductivity are computed. For the forced thickness vibrations of an infinite plate, the complex coupling factor K*, input admittance Y are derived and an explicit, approximate expression for K* is obtained in terms of material properties. Effects of the viscosity and conductivity on the resonance frequency, modes, admittance, attenuation coefficient, dynamic time constant, coupling factor, and quality factor are calculated and examined for quartz and ceramic barium titanate plates.

Second-order theories for extensional vibrations of piezoelectric crystal plates and strips.

An infinite system of two-dimensional (2-D) equations for piezoelectric plates with general symmetry and faces in contact with vacuum is derived from the 3-D equations of linear piezoelectricity in a manner similar to that of previous work, in which an infinite system of 2-D equations for plates with electroded faces was derived. By using a new truncation procedure, second-order equations for piezoelectric plates with faces in contact with either vacuums or electrodes are extracted from the aforementioned infinite systems of equations, respectively. The second-order equations for plates with or without electrodes are shown to predict accurate dispersion curves by comparing to the corresponding curves from the 3-D equations in a range up to the cut-off frequencies of the first symmetric thickness-stretch and the second symmetric thickness-shear modes without introducing any correction factors. Furthermore, a system of 1-D second-order equations for strips with rectangular cross section is deduced from the 2-D second-order equations by averaging variables across the narrow width of the plate. The present 1-D equations are used to study the extensional vibrations of barium titanate strips of finite length and narrow rectangular cross section. Predicted frequency spectra are compared with previously calculated results and experimental data.

Extensional, thickness-stretch and symmetric thickness-shear vibrations of piezoceramic disks.

A set of two-dimensional (2-D), second-order approximate equations for extensional, thickness-stretch and symmetric thickness-shear vibrations of piezoelectric ceramic plates with electroded faces is extracted from the infinite system of 2-D equations deduced previously. The new truncation procedure developed recently is used for it improves the accuracy of calculated dispersion curves. Closed-form solutions are obtained for free vibrations of circular disks of barium titanate. Dispersion curves calculated from the present approximate 2-D equations are compared with those obtained from the 3-D equations, and the predicted resonance frequencies are compared with experimental data. Both comparisons show good agreement without any corrections. The frequencies of the edge modes calculated from the present 2-D equations are very close to the experimental data. Furthermore, mode shapes at various frequencies are calculated in order to identify the frequency segments of the spectrum at which one of the coupled modes--i.e., the radial extension (R), edge mode (Eg), thickness-stretch (TSt), and symmetric thickness-shear (s.TSh)--is predominant.