Piezoelectric Half-space Research Articles

A penny-shaped interfacial crack between piezoelectric layer and elastic half-space

An interfacial penny-shaped crack between piezoelectric layer and elastic half-space subjected to mechanical and electric loads is investigated. Using Hankel transform technique, the mixed boundary value problem is reduced to a system of singular integral equations. The integral equations are further reduced to a system of algebraic equations with the aid of Jacobi polynomials. The stress intensity factor and energy release rate are determined. Numerical results reveal the effects of electric loadings and material parameters of composite on crack propagation and growth. The results seem useful for design of the piezoelectric composite structures and devices of high performance.

Effects of initial stress on the reflection and transmission waves at the interface between two piezoelectric half spaces

The effects of initial stress on the reflection and transmission waves at the interface between two piezoelectric half spaces are studied in this paper. First, the secular equations in the traverse isotropic piezoelectric half space are derived from the general dynamic equation with initial stress taken into consideration. Then, the interface conditions that displacement, stress, electric potential, and electric displacement are continuous across interface are required to be satisfied by three sets of coupled waves, namely, quasi-longitudinal wave, quasi-transverse wave and the electric–acoustic wave. The algebraic equations resulting from the interface conditions are solved to obtain the amplitude ratio of various waves and furthermore the energy reflection and transmission coefficients of various waves. The numerical results are shown graphically and the effects of initial stress are discussed.

Numerical simulations for the phase velocities and the electromechanical coupling factor of the Bleustein–Gulyaev waves in some piezoelectric smart materials

In this paper, the propagation behavior of the surface Bleustein–Gulyaev (B-G) waves in a piezoelectric layered half-space is investigated. The governing equations of the coupled waves are obtained. The boundary conditions are assumed in such a way that the displacements, shear stress, electric potential, and electric displacements are continuous across the interface between the layer and the substrate together with the traction-free boundary at the surface of the layer. The electrically open and short conditions at the surface are adopted to solve the problem. The phase velocity is numerically calculated for the electric open and short cases for different thicknesses of the layer and wave number. The phase velocity equation of the B-G wave is obtained by an analytical technique when a layered half-space with an identical piezoelectric layer and substrate but with opposite polarization is utilized. The electromechanical coupling factor in the layered piezoelectric structures is discussed. The results obtained in this paper will be very useful for the engineering application of B-G waves.

Propagation of a semi-infinite conducting crack in piezoelectric materials: Mode-I problem

In this paper, the mode-I transient response of a semi-infinite conducting crack propagating in a piezoelectric material with hexagonal symmetry under normal impact loading is investigated. The integral transform methods together with the Wiener–Hopf technique are used to solve the mixed boundary value problem under consideration. The solutions of the coupled fields are derived for two cases, i.e., generalized Rayleigh wave exists or not. The dynamic stress intensity factor and dynamic electric displacement intensity factor as well as their universal functions are obtained in a closed form. The numerical results for two universal functions are provided to illustrate the characteristics of dynamic crack propagation. It is found that the universal functions for the dynamic stress and electric displacement intensity factors vanish when the crack propagation speed reaches the generalized Rayleigh speed which is the propagation speed of surface wave in a piezoelectric half-space with metallized surface. It is noted that the electromechanical coupling coefficient has an important influence on the dynamic fracture characteristics.

Green’s function for an anisotropic piezoelectric half-space bonded to a thin piezoelectric layer

The Green’s function for an anisotropic piezoelectric half-space bonded to a thin piezoelectric layer subject to a generalized line force and a generalized line dislocation is presented. The thickness of a thin layer is assumed to be small compared with a reference length. Thus, the existence of the layer is replaced by effective boundary conditions to avoid finding solutions in the layer. Combining with the Stroh formalism gives explicit solutions in a more compact form.

The electromechanics of piezoresponse force microscopy for a transversely isotropic piezoelectric medium

Piezoresponse force microscopy (PFM) has emerged as one of the most powerful tools for characterizing and manipulating electromechanical responses of piezoelectric and ferroelectric materials at the nanoscale, yet the interpretation and quantitative analysis of PFM data remains difficult and is not well established. In this paper, we develop a rigorous analysis of PFM using the Hankel integral transform and the effective point charge model, which accounts for the electromechanical coupling of the transversely isotropic piezoelectric medium, the concentrated electric field induced by scanning probe microscopy tip, and the coupling of the electromechanical field at the interface of air and piezoelectric half-space. Using this method, the contact mode is first considered in comparison with the decoupled method, followed by a detailed analysis of the non-contact mode considering the effects of experimental conditions. We note that the decoupled method is only valid for materials with weak piezoelectricity, and the effective piezoelectric coefficient derived from PFM shows nonlinear dependence on the complete set of specimen electromechanical moduli, as well as on the experimental conditions. The analysis thus sheds considerable insight into PFM and could enable determination of intrinsic piezoelectric coefficients through PFM measurement using inverse calculations.

Reflection and Transmission Phenomena in Poroelastic Plate Sandwiched between Fluid Half Space and Porous Piezoelectric Half Space

The reflection and transmission of elastic waves in porous piezoelectric plate, overlying a porous piezoelectric half space and underlying a fluid half space, is studied. The constitutive and governing equations are formulated for porous piezoelectric materials. The expressions for the mechanical displacements, electric displacements, stresses, and electric potentials are derived for porous piezoelectric plate, porous piezoelectric half space, and fluid half space. The boundary conditions are described for the studied model. The behaviour of reflected and transmitted amplitude ratios relative to frequency, incident angle, thickness, and porosity is observed numerically. The impedance mismatching problem between the dense piezoelectric materials and the surrounding medium can be solved by the inclusion of porosity in dense piezoceramics.

Interfacial waves between piezoelectric and piezomagnetic half-spaces with magneto-electro-mechanical imperfect interface

We study the propagation of shear horizontal waves between the interface of piezoelectric and piezomagnetic half-spaces with a magneto-electro-mechanical imperfect contact. Mechanical, electrical and magnetical imperfections are modelled by means of a spring, a capacitor and a inductor, respectively. A general expression for the dispersion relation not reported previously in literature is given in an explicit form, with the diverse limit cases analysed in detail. In some of these limit cases, new expressions are also obtained which predict the existence of interfacial waves. In the other cases, when already reported results exist, a comparison with them is done. Some physical interpretations are derived from the limit cases. The influence of mechanical, electrical and magnetical imperfect contacts are shown in some numerical examples.

Full-field analysis of three dimensional piezoelectric layered half-space with point loading

Abstract In this study, a three dimensional transversely isotropic piezoelectric layered half-space subjected to concentrated loading and electric charge was analyzed. This paper provides an effective analytical methodology to obtain the explicit full-field solutions for this problem. Due to the dissimilar structures of the solutions, the applied loadings are separated into axis-symmetric loadings and non-symmetric loadings. By using the double Fourier transformation technique in conjunction with series expansion, the explicit closed-from solutions for displacements, stresses, and the electric field in the layered half-space are obtained. The analytical solutions are valid for loadings applied either in the interior or at the surface of the structure. It can be seen that the complete solutions obtained by the method used in this study comprise only the fundamental solutions of an infinite space with a concentrated loading. One of the fundamental solutions is the Green's function of an infinite space and the others are solutions induced to satisfy the boundary and interface conditions. The mathematical meaning of the method used in the study is in fact the image method or Mindlin's superposition method. Instead of finding image singularities manually, the intensity and locations of the image singularities are determined automatically from the mathematical technique used in this paper. Taking PZT materials as an example, the numerical computation for the full-field distributions of the displacement and electric field in the layered half-space medium subjected to a point force and a point charge was discussed.

Orientation effect on the Boussinesq indentation of a transversely isotropic piezoelectric material

Three-dimensional finite element analysis was used to study the effect of the angle between the loading direction and the axisymmetric direction on the indentation behavior of a transversely isotropic piezoelectric half-space by a cylindrical indenter of flat end. Two cases were considered in the analysis, which included (a) the indentation by an insulating indenter, and (b) the indentation by a conducting indenter. Both the indentation load and the indentation-induced potential were found to be proportional to the indentation depth. Using the simulation results and the analytical relationship for the indentation by a rigid, insulating indenter, semi-analytical relationships were developed between the indentation load and the indentation depth and between the indentation-induced potential on the indenter and the indentation depth for the conducting indenter, respectively. The proportionality between the indentation-induced potential and the indentation depth is only a function of the angle between the loading direction and the poling direction, independent of the type of indenters, which may be used to measure the relative direction of the loading axis to the axisymmetric axis of transversely piezoelectric materials from the indentation test.

Электроупругие поверхностные волны лява в слоистой структуре из пьезоэлектрического полупространства и диэлектрического слоя при наличии электрического экрана.

Электроупругие поверхностные волны лява в слоистой структуре из пьезоэлектрического полупространства и диэлектрического слоя при наличии электрического экрана.

Rayleigh Wave in a Rotating Initially Stressed Piezoelectric Half-Space

The governing equations of an initially stressed rotating piezoelectric medium are solved for surface wave solutions. The appropriate solutions in the half-space of the medium satisfy the required boundary conditions to obtain the frequency equation of Rayleigh wave for charge free as well as electrically shorted cases. The non-dimensional speed of the Rayleigh wave is computed numerically for particular examples of Lithium niobate and PZT-5H ceramics. The effects of rotation and initial stress are observed graphically on the non-dimensional speed of the Rayleigh wave.

Indentation responses of piezoelectric layered half-space

A method for solving a class of normal frictionless indentation problems in a multilayered piezoelectric half-space is proposed. The half-space is deformed by a rigid axisymmetric indenter that includes flat-ended cylinder, cone and sphere shapes. Three different interfacial models, i.e. perfectly bonded, frictionless contact and spring-like models, are investigated. Both insulating and conducting indenters are considered. The complete solution in the half-space is obtained by integrating the elastic and electric displacements due to a point load and a point charge with unknown distribution functions over the contact area. These distribution functions are to be determined by using the boundary conditions of the contact problems. The numerical results of the indentation responses of a piezoelectric thin film in contact with a piezoelectric half-space are given as an example of the proposed method.

The Piezoelectric and Piezomagnetic Effects on the Surface Wave Speed

the piezoelectric and piezomagnetic effects and the influence of short and open circuit on the surface wave speed are investigated in this paper. First, the elastic, piezoelectric and piezomagnetic coefficients in the considered ordinate system are obtained by Bonde transformation from that in the crystal axes ordinate system. Then, the equation which surface wave speed satisfies is derived from the free traction condition on the surface of piezoelectric and piezomagnetic half space with consideration of short and open circuit case. Some numerical examples are given and the piezoelectric and piezomagnetic effects and the influence of short and open circuit on the surface wave speed are shown graphically.

Interfacial waves between two piezoelectric half-spaces with electro-mechanical imperfect interface

We study the propagation of shear horizontal waves between the interface of two piezoelectric materials with an electro-mechanical imperfect contact. Mechanical and electrical imperfections are modeled by means of a spring and a capacitor, respectively. The corresponding mathematical expressions for the imperfect contact are given in this article. The system of differential equations for the waves in the considered half-space is derived and the associate solutions are found. A general expression for the dispersion relation, not reported previously in the literature, is given in an explicit form, with the diverse limit cases analyzed in detail. In some of these limit cases, new expressions are also obtained, which predict the existence of interfacial waves. In the other cases, where already reported results exist, a comparison with them is done. Some physical interpretations are derived from the limit cases. The influence of mechanical and electrical imperfect contacts are shown in some numerical examples.

Ultrasonic wave’s interaction at fluid-porous piezoelectric layered interface

The complete description of acoustic propagation in a multilayered system is of great interest in a variety of applications such as non-destructive evaluation and acoustic design and there is need for a flexible model that can describe the reflection and transmission of ultrasonic waves in these media. The reflection and transmission of ultrasonic waves from a fluid loaded porous piezoelectric layered structure is studied analytically. The layered structure is considered to be consisting of n number of layers of porous piezoelectric materials. Transfer matrix technique is used to study the layered materials. The analytical expressions for the reflected, transmitted, interaction energy ratios and surface impedance are obtained. The effects of frequency, porosity, angle of incidence, layer thickness and number of layers on the energy ratios and surface impedance are studied for different configurations of the layered materials. The results obtained are deduced for the poro-elastic and fluid loaded porous piezoelectric half space case, which are in agreement with earlier established results. A comparison of the results, obtained by alternate numerical techniques, is made.

Propagation of a mode-III interfacial crack in a piezoelectric–piezomagnetic bi-material

The transient response of a semi-infinite mode-III interfacial crack propagating between piezoelectric (PE) and piezomagnetic (PM) half spaces is investigated in this paper. The integral transform method together with the Wiener–Hopf and Cagniard–de Hoop techniques is used to solve the mixed boundary value problem under consideration. The existence of generalized Maerfeld–Tournois interfacial wave is discussed and the solutions of the coupled fields are derived for four different cases of bulk shear wave velocity. The dynamic intensity factors of stress, electric displacement and magnetic induction as well as energy release rate (ERR) are obtained in explicit forms. The numerical results of the universal functions and dimensionless ERR for several different material combinations are presented and discussed in details. It is found that the Bleustein–Gulyaev (generalized Maerfeld–Tournois) waves dominate the dynamic characteristics of the interfacial crack propagation in PE–PM bi-material.

SH Surface Waves in a Dielectric Layered Piezoelectric Half Space Loaded with a Liquid Layer

In this paper, we analyze the propagation of shear horizontal surface waves in a dielectric layer/piezoelectric substrate structure covered by a viscous liquid layer of finite thickness. The complex dispersion equation is derived. The numerical results show that the liquid layer thickness and dimensionless real-valued wave number have significant eff ects on the attenuation and phase velocity. In addition, it is found that in general the influence of the liquid layer viscosity on the phase velocity is opposite to that of the liquid layer density.

A Theoretical study of Propagation of Rayleigh surface waves in functionally graded piezoelectric (FGPM) half-space

This work presents a theoretical study of the propagation behavior of Rayleigh waves in a functionally graded piezoelectric material (FGPM) half-space. The influence of the graded variation of FGPM coefficients on the dispersion curves of Rayleigh waves is investigated for both electrically open and short conditions. The exponential gradient may be applied separately to mechanical and electrical properties. Examples have been given illustrating the use and capabilities of the method based on the recursive stiffness matrix method, in calculating SAW coupling factor (K2) versus crystal orientation, SAW phase velocity for particular frequency range, and field profile variations with depth. The effects of the gradient variation of material constants on the distribution of displacement, stresses and electric potential are presented and discussed in detail. It is note that this paper considers the plane problem of an anisotropic beam which the material composition varies continuously in the direction of the thickness. Some useful discussions and numerical approach are presented to show the significant influence of material inhomogeneity, and adopting a certain value of the inhomogeneity parameter α and applying suitable electric and mechanical loads can optimize the FGPM substrate structures. This will be of particular importance in modern engineering design of high-performance surface acoustic wave (SAW) devices.

Axisymmetric thermo-electro-elastic analysis of a piezoelectric half space

In this study, an analytical solution is presented for thermo-electro-elastic analysis of piezoelectric semi-infinite bodies. For this purpose, governing equations are derived for a transversely isotropic piezoelectric material under an axisymmetric thermo-electro-mechanical loading condition. A general closed-form analytical solution is presented for the complementary and particular parts of the components of the displacement vector and also for the electric potential function. Then, boundary conditions are imposed and in that case an explicit solution is obtained for piezoelectric semi-infinite bodies. Results show that when a piezoelectric half space is subjected to constant/ramp surface thermal loading the maximum absolute value of radial stress occurs at the surface of the body, whereas the maximum absolute value of stress in the normal direction of the half-space surface occurs not on the surface but somewhere near the surface of the body. Moreover, the peak values for these stress curves in the case of combined loadings are closest to the half-space surface and the farthest in the case of ramp decaying thermal loading. In contrast to the constant/ramp surface thermal loading, no clear maximum point for electric field distribution for the case of combined loadings can be seen in the normal direction to the half-space boundary.