Harry F. Tiersten

Electroelastic equations for electroded thin plates subject to large driving voltages

Existing rotationally invariant electroelastic equations are reduced to the case of large electric fields and small strain. These latter equations are specialized to the case of thin plates with completely electroded major surfaces, and it is shown that in this case the charge equation of electrostatics is satisfied trivially to lowest order. It is further shown that for the thin stress‐free polarized ferroelectric ceramic plate subject to large electric fields, the resulting equations readily account for experimental data in existence in the literature

A corrected modal representation of thickness vibrations in quartz plates and its influence on the transversely varying case

A modal representation of the thickness vibrations of rotated Y-cut quartz plates, which was used in the treatment of driven transversely varying thickness modes, is shown to be defective in certain respects. The differential equations and edge conditions for transversely varying thickness modes have been used in the accurate treatment of trapped energy resonators, monolithic crystal filters, and contoured quartz resonators, even though those defects were present. In this work those defects in the thickness solution are corrected along with the influence on the differential equations and edge conditions in the transversely varying case. The corrected modal representation shows that, because in practical applications to the above mentioned devices, the driving frequency is always near a thickness resonant frequency, essentially the same results will be obtained with the corrected representation as were obtained with the defective one, which explains why the results obtained with the defective equations were so accurate.

Equations for the control of the flexural vibrations of composite plates by partially electroded piezoelectric actuators

A system of 2D equations for the flexural vibrations of symmetric composite plates with both unelectroded and electroded portions of piezoelectric actuators attached to the top and bottom surfaces is obtained. The resulting system of equations is applied in the analysis of the cylindrical bending of a composite plate caused by voltages applied to partially electroded piezoelectric actuators. It is shown that no shearing stress is transferred across the interfaces between the electroded portions of the actuators and the composite plate and that an exponentially decaying shearing stress is transferred across the interfaces in the unelectroded regions of the actuators. The resulting finite shearing stress shows that when the electrodes end a short distance from the edge of the actuator, the undesirable singular shearing stress distribution that exists when the electrodes extend to the edge of the actuator is eliminated.

Transversely varying thickness modes in trapped energy resonators with shallow and beveled contours

The equation for transversely varying thickness modes in doubly rotated quartz resonators is applied in the analysis of contoured resonators with rectangular electrodes. The influence of both the contouring and the continuity conditions at the edges of the electrodes are included in the analysis. The steady‐state forced vibrations of contoured trapped energy resonators is treated and a lumped parameter representation of the admittance, which is valid in the vicinity of a resonance, is obtained. Calculated results are presented for a number of trapped energy resonators with shallow and beveled contours.

TRANSVERSELY VARYING THICKNESS MODES IN QUARTZ RESONATORS WITH BEVELED CYLINDRICAL EDGES

The equation for transversely varying thickness modes in doubly rotated quartz resonators is applied in the analysis of trapped energy resonators with beveled cylindrical edges. The coefficients appearing in the planar differential operator are written as a sum of a mean or isotropic part plus a deviation. Asymptotic eigensolutions for the nearby isotropic case are obtained for the cylindrical beveled resonator. The resonant frequencies for the actual anisotropic case are obtained from an equation for the perturbation in eigenfrequency from the isotropic solution. A lumped parameter representation of the admittance, which is valid in the vicinity of a resonance, is obtained. Calculated results are presented for a few beveled AT‐ and SC‐cut quartz resonators and the influence of the radius of curvature of the contour is exhibited.

On the necessity of including electrical conductivity in the description of piezoelectric fracture in real materials

Because no real material is a perfect insulator, but has some small electrical conductivity, under static circumstances the piezoelectric equations are not applicable to any real material and, hence, have no physical significance. On account of the existence of real current, when the time-dependence is sufficiently slow, the zero charge equation of electrostatics does not apply and must be replaced by the equation of the continuity of electric charge.

On the thickness expansion of the electric potential in the determination of two-dimensional equations for the vibration of electroded piezoelectric plates

In the derivation of two-dimensional equations for the vibration of piezoelectric plates from variational equations, expansions of the mechanical and electrical variables in the thickness coordinate are employed. If the major surfaces of the plate are electroded and the electric potential is expanded in functions of the thickness coordinate which do not vanish at the electrodes, the variations of the different orders of the expansion potentials are not independent because the electric potential must satisfy constraint conditions at the electrodes where it is independent of position. In this work, the electric potential is expanded in functions of the thickness coordinate which do not vanish at the surface electrodes and the constraint conditions are included by means of the method of Lagrange multipliers. The resulting piezoelectric plate equations are obtained along with an integral condition on the Lagrange multipliers over the electrodes, which results in the equation for the current through the electr...

Electroelastic equations describing slow hysteresis in polarized ferroelectric ceramic plates

Rotationally invariant electroelastic equations are extended to account for slow hysteretic effects in polarized ferroelectric ceramic plates by employing an internal variable in a thermodynamic state function. All material irreversibility is taken to be a consequence of the ferroelectric polarization-electric-field irreversibility. Since we are concerned with the slowest possible hysteresis, we ignore the evolution equation and take the known irreversible thickness polarization-electric-field saturation curve as the starting curve. After the general nonlinear three-dimensional equations are obtained, they are reduced to linear strain and then plane stress because all experimental data available to us are from strain measurements on a thin wide beam forced by voltages applied to very thin polarized ferroelectric ceramic actuators on the surfaces. The new irreversible material coefficients are obtained from data on one major hysteresis loop along with the coefficients in the saturation polarization-electri...

THEORY OF A FLOATING RANDOM-WALK ALGORITHM FOR SOLVING THE STEADY-STATE HEAT EQUATION IN COMPLEX, MATERIALLY INHOMOGENEOUS RECTILINEAR DOMAINS

Abstract We present the theory and preliminary numerical results for a new random-walk algorithm algorithm solves the steady-state heat equation subject to Dirichlet boundary conditions. Our emphasis is the analysis of geometrically complex domains made up of piecewise-rectilinear boundaries and material interfaces. This work is principally motivated by the semiconductor industry, specifically, their aggressive development of so-called multichip module (MCM) technology. We give a mathematical derivation of the surface Greens function for Laplaces equation over a square region. From it, we obtain an infinite multiple-integral series expansion yielding temperature at any space point in the actual heat-equation problem domain. A stochastic floating random-walk algorithm is then deduced from the integral series expansion. To determine the volumetric thermal distribution within the domain, we introduce a unique linear, bilinear, and trigonometric splining procedure. A numerical-verification study employing t...

Forced vibrations of the fundamental family of modes in AT-cut quartz strip resonators

Mindlin’s equations for the vibrations of elastic crystal plates are employed in the description of AT-cut quartz strip resonators. The electrically driven three-dimensional piezoelectric pure thickness solution is incorporated in the treatment. The driving voltage appearing in this thickness solution is included in the variational principle from which the plate equations are obtained. In this way the resulting Mindlin equations contain the driving voltage and hold for plates with small piezoelectric coupling. The equations are applied in the analysis of strip resonators. The eigensolutions are obtained by solving a sequence of one-dimensional problems that are defined by utilizing the results from the previous problem variationally. The driven solution is obtained by means of an expansion in the eigensolutions and a lumped parameter representation of the admittance, which is valid in the vicinity of a resonance, is obtained. Calculated results are presented for a range of geometries and the influence of the couplings is exhibited.