CuiYing Fan

Variational formulation of the Stevens-Tiersten equation and application in the analysis of rectangular trapped-energy quartz resonators

The two-dimensional scalar differential equation for transversely varying thickness modes in quartz crystal resonators operating with thickness-shear modes is formulated into variational form for trapped-energy resonators with both electroded and unelectroded regions. A theoretical analysis of rectangular trapped-energy resonators of singly rotated quartz is performed using the Ritz method based on the variational formulation. Free vibration resonant frequencies and modes are obtained. The results show the existence of trapped modes under the electrodes. The effects of various geometric and physical parameters on the trapped modes are examined. It is also found that the classical frequency prediction given by Tiersten and Smythe from an approximate analysis using the scalar differential equation has an inaccuracy on the order of 100 ppm for the fundamental mode, significant in resonator design.

Extended displacement discontinuity boundary integral equation and boundary element method for cracks in thermo-magneto-electro-elastic media

The extended displacement discontinuity boundary integral equation (EDDBIE) and boundary element method is developed for the analysis of planar cracks of arbitrary shape in the isotropic plane of three-dimensional (3D) transversely isotropic thermo-magneto-electro-elastic (TMEE) media. The extended displacement discontinuities (EDDs) include conventional displacement discontinuity, electric potential discontinuity, magnetic potential discontinuity, as well as temperature discontinuity across crack faces; correspondingly, the extended stresses represent conventional stress, electric displacement, magnetic induction and heat flux. Employing a Hankel transformation, the fundamental solutions for unit point EDDs in 3D transversely isotropic TMEE media are derived. The EDDBIEs for a planar crack of arbitrary shape in the isotropic plane of a 3D transversely isotropic TMEE medium are then established. Using the boundary integral equation method, the singularities of near-crack border fields are obtained and the extended stress field intensity factors are expressed in terms of the EDDs on crack faces. According to the analogy between the EDDBIEs for an isotropic thermoelastic material and TMEE medium, an analogical solution method for crack problems of a TMEE medium is proposed for coupled multi-field loadings. Employing constant triangular elements, the EDDBIEs are discretized and numerically solved. As an application, the problems of an elliptical crack subjected to combined mechanical-electric-magnetic-thermal loadings are investigated.

Variational analysis of thickness-shear vibrations of a quartz piezoelectric plate with two pairs of electrodes as an acoustic wave filter

We study thickness-shear vibrations of a piezoelectric plate of AT-cut quartz with two pairs of electrodes. The plate represents a monolithic, two-pole acoustic wave filter. The scalar differential equations by Tiersten and Smythe for thickness- shear vibrations of electroded and unelectroded quartz plates are employed. Based on the variational formulation of the scalar differential equations established in a previous paper and the Ritz method with trigonometric functions as basis functions, free vibration resonant frequencies and thickness-shear modes of the plate are obtained. For a structurally symmetric filter, the modes can be separated into symmetric and antisymmetric ones. Trapped modes with vibrations mainly under the electrodes are presented. The effects of the electrode inertia and the distance between the two pairs of electrodes are examined. It is also found that the classical frequency prediction given by Tiersten from an approximate analysis has an inaccuracy of tens of parts per million, significant in filter design and application.

Three-dimensional vertical cracks in magnetoelectroelastic media via the extended displacement discontinuity boundary integral equation method

In this article, the extended displacement discontinuity method is extended to study the fracture problem in a three-dimensional transversely isotropic magnetoelectroelastic medium with a planar crack vertical to the plane of isotropy. Considering the electric and magnetic fields in the crack cavity, and using the Somigliana identity along with the displacement discontinuity Green’s functions, the hyper-singular boundary integral equations for the unknown displacement discontinuities across the crack face are derived. The singularity features along the crack fronts are analyzed, and the extended field intensity factors are expressed in terms of the extended displacement discontinuities on the crack face. Numerical examples on the field intensity factors are finally calculated for a vertical square crack using the exact closed-form Green’s functions due to constant displacement discontinuities over a rectangular crack element, and some interesting features are observed, which are different from the case where the crack is located in the plane of isotropy. The influence of the electric and magnetic boundary conditions along the crack face on the field intensity factors is further studied.

Numerical method for nonlinear models of penny-shaped cracks in three-dimensional magnetoelectroelastic media

Green functions corresponding to uniformly distributed extended displacement discontinuities on an annular crack element in the isotropic plane of a three-dimensional transversely isotropic magnetoelectroelastic medium are derived. Using the obtained Green functions, an extended displacement discontinuity method is presented to analyze a penny-shaped crack under axisymmetric loadings. Using the electric and magnetic polarization saturation model and the electric and magnetic breakdown model, the electric and magnetic yielding zones, the extended displacement discontinuities, the extended stress intensity factors and the J-integral are numerically calculated. The accuracy and efficiency of the proposed method are demonstrated by comparing the numerical results with those obtained from analytical solutions.

Analysis of an interface crack of arbitrary shape in a three-dimensional transversely isotropic magnetoelectrothermoelastic bimaterial—part 1: Theoretical solution

ABSTRACT The extended displacement discontinuity boundary integral–differential equation method is developed for the analysis of an interface crack of arbitrary shape in a three-dimensional (3D), transversely isotropic magnetoelectrothermoelastic bimaterial. The extended displacement discontinuities (EDDs) include conventional displacement discontinuities, electric potential discontinuity, magnetic potential discontinuity as well as temperature discontinuity across the interface crack faces, correspondingly, while the extended stresses represent conventional mechanical stresses, electric displacement, magnetic induction, heat flux, etc. By virtue of the potential functions and Hankel transformation technique, the fundamental solutions for unit-point EDDs on the interface in a 3D transversely isotropic magnetoelectrothermoelastic bimaterial are derived, then the extended displacements and stresses are all obtained in terms of EDDs. An analysis method is proposed based on the analogy with the solution in an isotropic thermoelastic bimaterial. The singular indices and the singular behaviors of the near crack-tip fields are studied. The combined extended stress intensity factors for three new fracture modes are derived in terms of the EDDs and are compared with those in magnetoelectroelastic bimaterials.

Penny-shaped cracks in three-dimensional piezoelectric semiconductors via Green’s functions of extended displacement discontinuity

In this article, a penny-shaped crack in the isotropic plane of three-dimensional transversely isotropic piezoelectric semiconductors is analyzed via the displacement discontinuity boundary element method. The general solutions are derived based on the generalized Almansi’s theorem and the operator theory. Green’s functions are derived using the Hankel transformation under uniformly distributed extended displacement discontinuities (including displacement discontinuities, potential discontinuity, and electron density discontinuity) on a penny-shaped crack in the isotropic plane of three-dimensional transversely isotropic piezoelectric semiconductors. Using the extended displacement discontinuity boundary element method, penny-shaped cracks in transversely isotropic plane of three-dimensional piezoelectric semiconductors are studied, and the stress, electric displacement, and electric current intensity factors under uniform mechanical-electric-current loads applied on the penny-shaped crack surface are calculated. The fracture behaviors of piezoelectric semiconductors are studied.

A nonlinear bilayer beam model for an interfacial crack in dielectric bimaterials under mechanical/electrical loading

A bilayer beam model is extended to study the fracture behavior of dielectric interfacial cracks. In this model, a semi-infinite crack with an original opening value is oriented along the interface between two dielectric layers which are under mechanical/electrical loading. Taking into account the effect of the electrostatic traction on the interfacial crack, a nonlinear analytical solution is derived, along with also a developed finite element analysis method where a special constitutive equation for the capacitor element in ANSYS is utilized to simulate the electrostatic tractions. Both the analytical and numerical solutions predict the same results which further show that the elastic and dielectric mismatches can play a significant role in the interfacial cracking behavior under mechanical and electrical loading. Furthermore, the electrostatic tractions may cause hysteresis loops in the curve of crack opening versus applied mechanical displacement or versus applied electric voltage. An applied mechanical load is the driving force for the interfacial cracking, while an applied electric field retards it.

Analysis of arbitrarily shaped planar cracks in three-dimensional isotropic hygrothermoelastic media

ABSTRACT In the present article, a planar crack of arbitrary shape embedded in three-dimensional isotropic hygrothermoelastic media is investigated. Based on the general solutions and Hankel transform technique, the fundamental solutions for unit-point and extended displacement discontinuities (EDD; including the displacement discontinuities, moisture concentration discontinuity, and the temperature discontinuity) are derived. The EDD boundary integral equations for an arbitrarily shaped, planar crack in the hygrothermoelastic medium are established in terms of the EDD. Utilizing the boundary integral equation method, the singularities of near-crack front fields are analyzed, and the stress, moisture flux, and heat flux intensity factors are all derived in terms of the EDD. As a special case, the analytical solution for a penny-shaped crack under uniform combined loadings is presented. The EDD boundary element method is proposed for numerical simulation. The numerical result for a penny-shaped crack subjected to uniform mechanical–moisture–thermal loading is compared with the analytical solution to verify the correctness of the proposed method. Two coplanar elliptical cracks subjected to combined loadings are simulated as an application, and the influences of applied loadings and the ellipticity ratio are discussed.

Energy trapping of thickness-shear modes in inverted-mesa AT-cut quartz piezoelectric resonators

abstract We studied thickness-shear vibration frequencies and modes in an inverted-mesa AT-cut quartz plate piezoelectric resonator thinner in the center and thicker near the edges. It is electroded in the central part and unelectroded near the edges. A theoretical analysis was performed using the scalar differential equations by Stevens and Tiersten for thickness-shear vibrations of electroded and unelectroded quartz plates. Based on the variational formulation of the scalar differential equations established in a previous paper and the variation-based Ritz method with trigonometric functions as basis functions, free vibration resonant frequencies and the corresponding thickness-shear modes were obtained. It was found that the electrodes tend to trap the vibration in the central part of the resonator (energy trapping) but the curvature of the resonator surface has an opposite effect. The trapping is sensitive to the electrode thickness, the electrode length, and the curvature of the resonator surface. Hence accurate design is needed to achieve proper energy trapping in inverted-mesa resonators.