MingHao Zhao

Analysis of a penny-shaped crack in a magneto-electro-elastic medium

The solution for an ellipsoidal cavity in an infinite transversely isotropic magneto-electro-elastic medium under remotely applied axisymmetric, combined mechanical–electric–magnetic loading is derived. The exact solution for a penny-shaped crack is obtained by letting the minor axis of the ellipsoidal cavity approach zero. The results demonstrate that the stress intensity factor depends only on the applied mechanical loading, but the electric displacement intensity factor and the magnetic induction intensity factor depend on the applied combined loading, as well as the two ratios of β/α and γ/α. Parameter α is the ratio of the minor axis to the major axis of the ellipsoidal cavity, β is the ratio of the dielectric constant of the material in the cavity to the effective dielectric constant of the magneto-electro-elastic medium, and γ is the ratio of the magnetic permeability of the material in the cavity to the effective magnetic permeability of the magneto-electro-elastic medium. The two ratios characterize the permeability of the crack to electric and magnetic fields. Several limiting cases for β/α and γ/α are studied. A self-consistent method is adopted to determine the real crack opening α under the combined mechanical–electric–magnetic loading. The stress, electric displacement and magnetic induction intensity factors of a penny-shaped crack in BaTiO3–CoFe2O4 composites are calculated for different volume fractions and different applied combined loadings.

Failure behavior and failure criterion of conductive cracks (deep notches) in thermally depoled PZT-4 ceramics

In the present work, we experimentally and theoretically studied the failure behavior of electrically conductive cracks (deep notches) in thermally depoled PZT-4 ceramics, which were macroscopically dielectric, under mechanical and/or electrical loading. When the critical stress intensity factor was normalized by the critical stress intensity factor under purely mechanical loading and the critical electric intensity factor was normalized by the critical electric intensity factor under purely electric loading, the experimental results revealed that the failure behavior of the conductive cracks in the ceramics was described by a quadratic function of the normalized electric intensity factor versus the normalized stress intensity factor. Accordingly, we proposed herein a charge-free zone model to predict the failure behavior, by treating dielectric ceramics as mechanically brittle and electrically ductile materials. The charge-free zone model resulted in a failure criterion, which agreed perfectly with the experimental results.

Equilibrium depth and spacing of cracks in a tensile residual stressed thin film deposited on a brittle substrate

The depth and spacing of cracks in a tensile residual stressed thin film bonded on a brittle substrate are analyzed thermodynamically using the minimum energy theorem on the basis that the film has the same mechanical properties as the substrate. The results show that the cracks penetrate into the substrate. Simple and approximate relationships between three dimensionless parameters, i.e., the normalized crack depth and spacing, and the cracking resistance number, are derived, which determine the fracture behavior of the film.

Indentation fracture and indentation delamination in ZnO film/Si substrate systems

ZnO films with thicknesses ranging from 0.202 to 1.535 µm were deposited by using the magnetron sputtering technique on Si (100) substrates 525 µm thick. Then, Vickers indentation tests were carried out on the ZnO/Si systems at room temperature, in which the applied load varied from 10 mN to 2.0 N. The experimental results show that only indentation-induced radial cracking occurred in the systems with film thicknesses equal to and thinner than 0.554 µm, from which the residual stress in the films was extracted to be 387 MPa in compression. For the systems with film thicknesses equal to and thicker than 0.832 µm, only indentation-induced delamination occurred when indentation loads were low. Under high indentation loads, radial cracking concurrently occurred with delamination. The radial cracks were invisible at the film surfaces because the crack length was smaller than the delamination size. The critical film thickness for indentation-induced delamination was found to be around 0.7 µm for the ZnO/Si systems. Combining the composite hardness models with the indentation-induced delamination model, we developed a method to determine the interfacial fracture energy between a film and its substrate. The novel method is particularly useful for indentation equipment without any displacement measurement devices. Using the new method, we extracted the interfacial fracture energy to be about 12.2 J m−2 and from 9.2 to 11.7 J m−2 for the cases without and with buckling respectively of delaminated films. Consequently, the pure mode I interfacial fracture energy was calculated to be 10.4 J m−2 for the ZnO/Si systems.

Interaction of a piezoelectric screw dislocation with an insulating crack

Abstract The electrical and mechanical fields induced by a piezoelectric screw dislocation near an electrically insulating elliptical cavity in a piezoelectric material are derived through the image dislocation approach by considering the electric field inside the cavity. When the cavity is reduced to a crack, three outcomes are possible. The three outcomes correspond to three different electrical boundary conditions along the crack faces, depending on the α/β; ratio, where α; is the ratio of the minor semiaxis to the major semiaxis of the ellipse and β; is the ratio of the dielectric constant of the cavity to the effective dielectric constant of the piezoelectric material. The crack is electrically impermeable when α/β; → ∞ the crack becomes electrically permeable as α/β; → 0. Since the minimum of the dielectric constant has a finite non-zero value and a real crack also has a non-zero width, the α/β; ratio will generally have a finite non-zero value, resulting in a semi-impermeable crack. Furthermore, the difference in the electric boundary conditions leads to great differences in the image force acting on the dislocation, in the intensity factors and in the J integral for crack propagation induced by the dislocation.

Effects of electric fields on the bending behavior of PZT-5H piezoelectric laminates

This paper investigates the bending behavior of piezoelectric laminates under combined mechanical and electrical loads. The laminates have a PZT-5H ceramic core sandwiched between two graphite/epoxy plates. Three-point bending tests and in situ acoustic emission measurements were conducted on the PZT-5H sandwich laminates with a sustained applied electric field. The results show that the PZT-5H core fractures first and then delaminaton occurs along the tensile stressed interface between the PZT ceramic and a graphite/epoxy layer. Finite element analysis was performed to analyze the stresses in the sandwich structure and the energy release rate for interface delamination under combined mechanical and electrical loads. Consequently, both the bending strength of the PZT core and the interface toughness were evaluated from the experimental data. Although the experimental data scatter widely, the electric field, either positive or negative, reduces the average fracture strength of the PZT core. The average interface toughness decreases on increasing the magnitude of a negative electric field, while it is increased by a positive electric field.

Electro-elastic analysis of piezoelectric laminated plates

Based on the Kirchhoff hypothesis of normal-remain-normal, the present work analyses piezoelectric laminated plates, wherein poled piezoelectric laminae are transversely isotropic and function as actuators. A quadric electric field is induced inside a piezoelectric lamina under a given applied voltage and mechanical bending. The governing equations for the piezoelectric laminated plate derived from the principle of virtual work in terms of the electric enthalpy have the same forms as those for a conventional composite laminated plate. We use rectangular sandwich plates of Al/PZT/Al and PZT/Al/PZT with four simply supported edges to demonstrate the prediction of the maximum bending stress in the PZT layer. The analytic solutions are verified by three-dimensional finite element analysis.

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.