Propagation of the Anisimkin Jr.' and quasi-longitudinal acoustic plate modes in low-symmetry crystals of arbitrary orientation

The Anisimkin Jr. (AN) acoustic plate mode having dominant and depth-independent longitudinal displacement (u(1) > u(2), u(3); u(1) ≈ constant) is numerically found in tetragonal 4mm Li(2)B(4)O(7) crystal with one of the low-symmetry orientations (Euler angles 89°, 37°, 104°), as an example. The quasi-longitudinal (QL) modes with dominant and depth-dependent longitudinal displacement (u(1) > u(2), u(3); u(1) ≠ constant) are experimentally detected along several propagation directions Θ in 128°y-LiNbO(3) plate, where Θ = 0°, 30°, 60°, and 90° with respect to the x-axis. Compared with more symmetrical plate materials and orientations, the displacement profiles of the AN and QL modes in lower-symmetry counterparts are qualitatively the same, but their phase profiles are more complicated. Moreover, like any acoustic wave, all plate modes in anisotropic crystals suffer from beam steering, in general. The power flow angles of the modes propagating in a fixed direction are different and depend on the mode order n.     

Peculiarities of the Anisimkin Jr.' plate modes in LiNbO3 and Te single crystals

Quasi-longitudinal (QL) Anisimkin Jr.' modes discovered recently in quartz plates are found now in 2 other piezoelectric crystals, belonging to trigonal symmetry. In 128°Y,X+90°-LiNbO3, 2 different QL modes may propagate simultaneously for small plate thickness h/λ = 0 to 0.06 (h is thickness, λ is wavelength). The velocity of the 1st mode, v1, is close to the longitudinal bulk wave velocity vL. It is varied with h/λ and piezoelectrically stiffened (maximum Kn² = 39% at h/λ = 0.08). The velocity of the 2nd QL mode, v2, is close to the shear-horizontal bulk wave velocity vQSH, not varied with h/λ and not stiffened (Kn² = 0). On the contrary, 210°Y,X-Te crystal supports only one QL-mode, but it is unusually wideranging and low-dispersive: the mode exists for all h/λ from 0 to 2.5 with velocity vn almost permanent and equal to vL in the whole range. This mode is piezoelectrically stiffened (maximum Kn² = 2% at h/λ = 0.13). The variety of the Anisimkin Jr.' modes in different crystals makes them attractive for liquid sensors, where the amount of suitable waves is very restricted.     

Properties of shear horizontal acoustic plate modes in BT-cut quartz

Properties of shear horizontal acoustic plate modes (SHAPMs) in BT-cut quartz were calculated and measured. A delay line with a long interdigital transducer, deposited on -50.5°YX90°-oriented quartz plate, was used for the measurements. For one of the SHAPMs, at a frequency of about 100.4 MHz, insertion loss, turnover temperature, and quadratic temperature coefficient of frequency of about 10 dB, 15°C, and -30 ppb/(°C)(2) in air, respectively, were obtained. Using water and glycerin solutions, insertion loss changes against dynamic viscosity were measured for this mode. In a viscosity range from about 1 mPa·s to 1000 mPa·s, an insertion loss change of about 14 dB was obtained.     

On the accuracy of Mindlin plate predictions for the frequency-temperature behavior of resonant modes in AT- and SC-cut quartz plates

The frequency spectra of resonant modes in AT- and SC-cut quartz plates and their frequency-temperature behavior were studied using Mindlin first- and third-order plate equations. Both straight-crested wave solutions and two-dimensional plate solutions were studied. The first-order Mindlin plate theory with shear correction factors was previously found to yield inaccurate frequency spectra of the modes in the vicinity of the fundamental thickness-shear frequency. The third-order Mindlin plate equations without correction factors, on the other hand, predict well the frequency spectrum in the same vicinity. In general, the frequency-temperature curves of the fundamental thickness-shear obtained from the first-order Mindlin plate theory are sufficiently different from those of the third-order Mindlin plate theory that they raise concerns. The least accurately predicted mode of vibration is the flexure mode, which results in discrepancies in its frequency-temperature behavior. The accuracy of other modes of vibrations depends on the degree of couplings with the flexure mode. Mindlin first-order plate theory with only the shear correction factors is not sufficiently accurate for high frequency crystal vibrations at the fundamental thickness-shear frequency. Comparison with measured resonant frequencies and frequency-temperature results on an AT-cut quartz plate shows that the third-order plate theory is more accurate than the first-order plate theory; this is especially true for the technically important fundamental thickness shear mode in the AT-cut quartz plate.     

Orientation of doubly rotated quartz plates

A derivation from classical spherical trigonometry of equations to compute the orientation of doubly-rotated quartz blanks from Bragg X-ray data is discussed. These are usually derived by compact and efficient vector methods, which are reviewed briefly. They are solved by generating a quadratic equation with numerical coefficients. Two methods exist for performing the computation from measurements against two planes: a direct solution by a quadratic equation and a process of convergent iteration. Both have a spurious solution. Measurement against three lattice planes yields a set of three linear equations the solution of which is an unambiguous result.     

Study of the bending modes in circular quartz resonators

An experimental and theoretical study of bending modes in a partially electroded circular piezoelectric quartz (AT-cut) with free edge is presented. The quartz is excited by a voltage pulse applied on the electrodes, and its surface is scanned by a laser vibrometer that measures the out-of-plane displacements. The classical theory of bending of thin disks is used to describe the flexural modes at frequencies lower than the first thickness shear resonance (6 MHz). A fairly good agreement is found between experimental and theoretical results for the forced mode shapes and for the resonance frequencies. However, it appears that the two springs used to maintain the disk in position introduce extra clamping conditions. Several source shapes were studied, among which a collection of an arbitrary number of forces is particularly useful. The two-dimensional wavenumber representation shows the presence of anisotropy related to the crystallographic axes at higher frequencies, which is not predicted by the model. The experimental phase velocities are compared to those given by the classical theory of disks and to those of Lamb A(0) mode. This study confirms the correspondence at low frequencies between the A(0) mode and the bending eigenmodes of a disk with finite size.     

Orientation effects in chemical etching of quartz plates

The development of etch patterns of z-, y- and x-cut quartz plate is studied as a function of the specimen orientation. Plates of different cuts exhibit typical etch patterns satisfying either digonal or trigonal symmetry of -quartz crystal. The changes of surface profilometry traces with depth of etch are also found to depend mainly on orientation dependence. The experimental data can be interpreted satisfactorily and consistently in terms of the structural reaction mechanism proposed by Ernsberger and in terms of the stability criterion. Some progress in the prediction of changes of etch figures with prolonged dissolution can thus be made by using some of the information reported in this paper.     

An analysis of nonlinear vibrations of coupled thickness-shear and flexural modes of quartz crystal plates with the homotopy analysis method

We investigated the nonlinear vibrations of the coupled thickness-shear and flexural modes of quartz crystal plates with the nonlinear Mindlin plate equations, taking into consideration the kinematic and material nonlinearities. The nonlinear Mindlin plate equations for strongly coupled thickness- shear and flexural modes have been established by following Mindlin with the nonlinear constitutive relations and approximation procedures. Based on the long thickness-shear wave approximation and aided by corresponding linear solutions, the nonlinear equation of thickness-shear vibrations of quartz crystal plate has been solved by the combination of the Galerkin and homotopy analysis methods. The amplitude frequency relation we obtained showed that the nonlinear frequency of thickness-shear vibrations depends on the vibration amplitude, thickness, and length of plate, which is significantly different from the linear case. Numerical results from this study also indicated that neither kinematic nor material nonlinearities are the main factors in frequency shifts and performance fluctuation of the quartz crystal resonators we have observed. These efforts will result in applicable solution techniques for further studies of nonlinear effects of quartz plates under bias fields for the precise analysis and design of quartz crystal resonators.     

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 2D equations for piezoelectric crystal plates with thickness-graded material properties, which is deduced from the 3D 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 2D equations as well as the 3D 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 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 Ω vs. a/b_q (length-to-thickness ratio of the quartz) are computed for plates with gold and aluminum electrodes and different R ratios. The spectrum of Ω vs. R is computed for plates with aluminum electrodes and a given a/b_q 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     

Crack bifurcation modes in composite plates under impact

Cracked plates with bonded elastic circular inclusions inside a matrix were subjected to an impact tensile loading. The influence of the inclusion on the propagation of an initial edge-crack was discussed in this paper. Two types of specimens were tested, that is with hard or soft inclusions surrounded by soft or hard matrices respectively. It has been shown that bi-, or tri-furcations of the propagating crack have persistently occured at the region of mesophase between matrix and circular inclusions. The phenomenon of splitting of the initial crack was common in both types of specimens.For the definition of the prevailing stress field, the dynamic stress intensity factors were evaluated at different positions of the crack, as well as the crack-velocities along the main crack or its branches during their propagation. It was observed that the crack initiation in soft plates with hard inclusions was followed by the emission of Rayleigh waves, which contributed to the mode of fracture of the composite. Isochromatic patterns were also used to simulate the stress-field of the propagating cracks against the inclusions at distinct time-instants where the propagating crack was frozen at this instant and static photoelastic analysis was applied. These approximations gave satisfactory results with the cracks advancing inside the composite plates.

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Résumé On a soumis à des tractions par impact des plaques fissurées comportant des inclusions élastiques circulaires ancrées dans une matrice. On discute ici de l'influence de l'inclusion sur la propagation d'une fissure de bord initiale. On a essayé deux types d'éprouvettes, comportant respectivement des inclusions dures ou tendres, noyées dans une matrice tendre ou dure. On montre que, systématiquement, la fissure en propagation fait état de bifurcations dans la zone de mésophase située entre la matrice et les inclusions circulaires. Dans les deux types d'éprouvettes, on a constaté qu'il était courant que la fissure initiale subisse un phénomène de séparation. Pour définir le champ de contraintes déterminant, on a évalué les facteurs dynamiques d'intensité des contraintes à différentes positions de la fissure, ainsi que les vitesses de la fissuration suivant la fissure principale suivant ses branches au cours de la propagation. On observe que l'amorçage de la fissure dans les plaques tendres comportant des inclusions dures est suivi d'une émission d'ondes de Rayleigh qui contribuent à déterminer le mode de rupture du composite. On a également utilisé des tracés isochromatiques pour simuler le champ de contrainte, en figeant la fissure en cours de propagation et en procédant à une analyse photo-élastique. Ces approximations ont conduit à des résultats satisfaisants dans le cas de fissures qui s'avancent dans des plaques composites.

     

加速度对Y型石英谐振器频率变化的影响

1引言 本文研究了Y型石英谐振器在不同支撑情况下产生的随着加速度方向变化的谐振频率的变化.首先利用石英谐振器在受到加速度力之后的应力解析解得出了不同支撑时板上每一点的应力分量以及应变分量,把所得结果带入二维的板振动方程后,再利用微小振动原理,最终得出了我们需要的频率变化量.     

Frequency-temperature behavior of spurious vibrations of rectangular AT-cut quartz plates

We evaluate the frequency-temperature behavior of spurious modes in a rectangular AT-cut quartz plate resonator based on three-dimensional linear equations. The elastic constants and three geometrical dimensions of the resonator are defined in terms of cubic polynomials of a temperature change. Assuming that the resonator holds its rectangular plate shape irrespective of temperature, we can determine the relationship between frequency and the dimensions of the resonator for a given temperature using the previous technique. We compare the calculated results with our own experimental data, and show that agreement between the calculated and observed data is excellent.     

A numerical approach to the analysis of failure modes in anisotropic plates

This study is motivated by the attempt to characterize failure modes of silicon chips commonly used in electronic industries. Previous experimental investigations provided the failure probability of dies made of a single-crystal and produced a large variety of crack patterns, but were not able to elucidate the link between defect distributions and crack initiation and propagation. To get some insight in the fracture activation and propagation mechanisms, we resort to finite element analyses and adopt an explicit methodology for crack tracking, based on the self-adaptive insertion of cohesive elements into a coherent mesh of solid elements. Finite kinematics material models with anisotropic features for both bulk and cohesive surfaces are employed to describe the behavior of single-crystal silicon plates undergoing a particular bending test up to failure. The cohesive model adopted in the calculation is fully anisotropic and newly formulated to accomplish the present study. Numerical simulations considering different material properties were able to ascertain the effects of particular flaws on failure modes of brittle silicon plates.     

Membrane analogy of the Stevens-Tiersten equation for essentially thickness modes in plate quartz resonators

We show that the well-known Stevens-Tiersten equation for essentially thickness modes in doubly rotated quartz resonators is mathematically analogous to the equation governing the transverse motion of a pre-stretched elastic membrane resting on an elastic foundation. The implication of this analogy is that commercial finite element software for structural analysis can be used to solve the Stevens-Tiersten equation for resonators.     

Five-mode frequency spectra of x3-dependent modes in AT-cut quartz resonators

We study straight-crested waves and vibration modes with variations along the x(3) direction only in an AT-cut quartz plate resonator near the operating frequency of the fundamental thickness-shear mode. Mindlin's two-dimensional equations for anisotropic crystal plates are used. Dispersion relations and frequency spectra of the five relevant waves are obtained. It is found that, to avoid unwanted couplings between the resonator operating mode and other undesirable modes, in addition to certain known values of the plate length/thickness ratio that need to be avoided, an additional series of discrete values of the plate length/thickness ratio also must be excluded.     

The viscosity of thin water films between two quartz glass plates

Summary An apparatus for the determination of the viscosity coefficient of water in the neighbourhood of a quartz glass surface is described. The law of Stefan-Reynolds is applied in a special modification. The relative viscosity coefficient of water between two quartz glass plates having a distance smaller than 100 ? lies beyond 10 for small shearing stresses. Increasing shearing stresses lead to smaller values of the relative viscosity coefficient of such water zone, so that a non-Newtonian behaviour of flowing water in very thin layers can be accepted.

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Résumé On décrit un appareil pour la détermination du coefficient de viscosité de l'eau au voisinage d'une surface de quartz transparent. On applique ici une variante de la loi de Stefan-Reynolds. Le coefficient de viscosité relatif de l'eau entre deux plaques de quartz transparent maintenues à seulement 100 ? l'une de l'autre est de plus de 10 pour les faibles contraintes de cisaillement. Des contraintes de cisaillement croissantes déterminent des valeurs plus faibles du coefficient de viscosité relative, et l'on peut donc admettre que l'eau, sous forme de ces couches très minces, se caractérise par un écoulement non newtonien.

     

Electromechanical coupling to Lamb and shear-horizontal modes in piezoelectric plates

Recent theoretical studies and experiments have been shown that interdigital transducers can couple strongly to plate modes in piezoelectric materials and in piezoelectric-on-nonpiezoelectric composite membranes. The calculated velocity dispersion and electromechanical coupling factors for plate modes in representative piezoelectric materials are described. The frequency dependence of velocity and electromechanical coupling factors are given, under different metallization conditions, for generalized stiffened-Lamb, pure stiffened-Lamb, and stiffened-shear (shear-horizontal) modes, for various plate orientations in lithium niobate, lithium tantalate, quartz, bismuth germanium oxide, and zinc oxide. For lithium niobate, electromechanical-coupling values as high as 15% are found under narrowband bandpass conditions, and 5% under wideband low-pass conditions. For lithium tantalate, bismuth germanium oxide, coupling values of 0.5, 2, and 4% are obtained. For quartz with its weaker piezoelectricity, the coupling is still made smaller.