Piezoelectric Half-space Research Articles

The Rotation Effect on Shear Horizontal Surface Wave in a Piezoelectric Semi-Space Covered by a Semiconductor Film

In this study, a mathematical model of the rotation effect on Shear Horizontally wave (SH-wave) propagation in a piezoelectric semi-space covered by a semiconductor film is investigated. The semiconducting layer is rotating with a constant angular velocity and the interface between the piezoelectric substrate and the semiconductor layer is imperfectly bonded. Furthermore, the surfaces of the bilayer system assumed free of traction and electrically shorted or open. The governing equations of the dynamical displacement and electrical potential function under the effect of rotation are driven by solving the coupled electromechanical field equations of the piezoelectric half-space and the semiconductor film. In addition, the exact frequency equations of SH waves are derived. Next, the numerical examples are considered to clarify the effects of rotation and electromagnetic boundary conditions for the different values of the film thickness and wave number on the dispersion behaviors. Finally, the effect of rotation on the frequency equation is investigated in detail for piezoelectric material PZT-5H and the semiconductor silicon. The obtained results provide a predictable and theoretical basis for applications of piezoelectric and semiconductor structures to surface acoustic wave equipments.

Scattering of SH-Wave by a Circular Inclusion Near the Interfacial Cracks in the Piezoelectric Bi-Material Half-Space

AbstractWith the aid of the Green's function method and complex function method, the scattering problem of SH-wave by a circular inclusion near the two symmetrically permeable interfacial cracks in the piezoelectric bi-material half -space is considered to obtain the steady state response. Firstly, by means of the image method, the essential function of Green's function is constructed, which satisfies the stress free and electric insulation conditions on the horizontal boundaries in a right-angle space including a circular inclusion and bearing a harmonic out-plane line source force on the vertical boundary. Secondly, the bi-material media is divided into two parts along the vertical boundary. According to continuity condition, the first kind of Fredholm integral equations containing undetermined anti-plane forces are established by “the conjunction method” and “the crack-division technology”, then the integral equations are reduced to the algebraic equations including finite items by effective truncation. Finally, the dynamic stress concentration factor around the edge of circular inclusion and dynamic stress intensity factor at the crack tip are calculated, then the influences of the frequency of incident wave, the length of crack, the position of the crack, the position of circular inclusion, etc. on the dynamic stress concentration factor and dynamic stress intensity factor are discussed.

Modeling of contact interaction of piezoelectric half-space and elastic isotropic base with surface groove of circular section

The mathematical model for the contact interaction under the compression of an electroelastic half-space with the elastic isotropic base containing a sloping groove of a circular section was developed. The model takes in account the connectedness of the electroelastic equations system. Two algorithms for solving the problem were proposed, which were based on the representation of solutions of the electroelasticity equations. The analytical problem solution was found, geometrical parameters of clearance between the electroelasticity half-space and deformable base under the known compression were obtained. The influence of connectedness of force and electricity fields on contact parameters of bodies was investigated.

The Frequency Equations for Shear Horizontal Waves in Semiconductor/Piezoelectric Structures Under the Influence of Initial Stress

In this paper, we investigated analytically the frequency equations for shear horizontal wave propagation in a piezoelectric half space covered by a semiconductor film with initial stress effect. The semiconducting layer is influenced by initial stress and the interface between the piezoelectric substrate and the semiconductor layer. The governing equations of the mechanical displacement and electrical potential function under the effect of initial stress are obtained by solving the coupled electromechanical field equations of the piezoelectric half-space and the semiconductor film. Next, the numerical examples are presented to illustrate the influence of initial stress and electromagnetic boundary conditions for the different values of the film thickness and wave number. Furthermore, we studied in more details the effect of initial stress on the frequency equation for piezoelectric Barium Titanate (BaTiO3) and semiconductor silicon. The obtained results provide a predictable and theoretical basis for applications of piezoelectric and semiconductor composites to acoustic wave devices.

Wave propagation through an inhomogeneous slab sandwiched by the piezoelectric and the piezomagnetic half spaces

Wave propagation through a gradient slab sandwiched by the piezoelectric and the piezomagnetic half spaces are studied in this paper. First, the secular equations in the transverse isotropic piezoelectric/piezomagnetic half spaces are derived from the general dynamic equation. Then, the state vectors at piezoelectric and piezomagnetic half spaces are related to the amplitudes of various possible waves. The state transfer equation of the functionally graded slab is derived from the equations of motion by the reduction of order, and the transfer matrix of the functionally gradient slab is obtained by solving the state transfer equation with the spatial-varying coefficient. Finally, the continuous interface conditions are used to lead to the resultant algebraic equations. The algebraic equations are solved to obtain the amplitude ratios of various waves which are further used to obtain the energy reflection and transmission coefficients of various waves. The numerical results are shown graphically and are validated by the energy conservation law. Based on the numerical results on the fives of gradient profiles, the influences of the graded slab on the wave propagation are discussed. It is found that the reflection and transmission coefficients are obviously dependent upon the gradient profile. The various surface waves are more sensitive to the gradient profile than the bulk waves.

Indentation response of piezoelectric films under frictional contact

Piezoelectric materials are available in a variety of shapes and forms in smart systems. One of the most common shapes for sensor or actuator applications is the film form. To gain a better understanding of the indentation behavior of these systems, this work analyzes piezoelectric films under different 3D frictional contact boundary conditions. For this purpose, the boundary element formulation presented by authors in Rodríguez-Tembleque, Buroni, and Sáez (2015) is used for modeling piezoelectric finitely thick and thin films under orthotropic frictional indentation conditions, including tangential loads. The formulation has been applied to analyze the influence of friction and tangential loads on the electromechanical response of finitely thick piezoelectric films, ranging from a piezoelectric half space configuration, where the contact radius is much smaller than the thickness of the film, to a thin film configuration. Results reveal that these variables have to be considered to study the indentation response of these advanced systems. In other case, we could over- or underestimate the piezoelectric response and its distribution over the contact zone.

Elastodynamic response for the multi-layered transversely isotropic piezoelectric solid subjected to time-harmonic loads

The precise integration algorithm (PIA) and the approach of dual vector form are utilized to explore the elastodynamic axisymmetric response of the multi-layered transversely isotropic piezoelectric medium. The planes of transverse isotropy are assumed to be parallel to the horizontal surface of the layered system. The elastic solid is under the action of axisymmetric mechanical or electrical time-harmonic loads prescribed either at the external surface or in the interior of the stratified medium. There is no limit of the thickness and number of layers to be considered. Based on the dual vector form and the Hankel integral transform, the governing equations of the cross-anisotropic piezoelectric material are conveniently turned into the standard ordinary differential matrix equation. The PIA is introduced to evaluate the first-order ordinary differential matrix equation with specified two-end boundary conditions. As PIA is a highly accurate method, any desired accuracy of dynamic piezoelectric solutions can be accomplished. In addition, the dual vector form of motion equation facilitates the combination of two neighboring layers into a new one and computational efforts could be reduced to a great extent. The true piezoelectric field is obtained by taking inversion of the Hankel integral transform. Finally, a comparison with the numerical solutions for stratified piezoelectric half-space is made to confirm the accuracy of the proposed procedure. Meanwhile, some numerical examples are also included to discuss the influence of loading positions, types of external forces, stratified characters and weak and thin interlayer on the dynamic response of the layered piezoelectric medium.

Effective behavior of a microscopically damaged interface between a layer and a half-space occupied by dissimilar piezoelectric media under antiplane deformations

The present paper examines the effective macroscopic behavior of a microscopically damaged interface between an infinitely long piezoelectric layer and a piezoelectric half-space under antiplane deformation. The interface is modeled as containing a periodic array of micro-cracks. The lengths and the positions of the micro-cracks on a period interval of the interface are randomly generated. The micro-statistical model is formulated in terms of hypersingular integral equations and used to investigate in detail the influences of the material constants of the piezoelectric layer and the half-space and the width of the layer on the effective properties of the interface.

Rayleigh wave on the half-space with a gradient piezoelectric layer and imperfect interface

In this study, we consider the propagation of a Rayleigh wave on an anisotropic half-space with a piezoelectric gradient covering layer and imperfect interface. First, the state transfer equation is derived from the governing equations and constitutive relations. The transfer matrix of the state vector is then obtained by solving the state transfer equation and the stiffness matrix is obtained. The total surface stiffness matrix is obtained by combining the transfer matrices and the stiffness matrices of the piezoelectric half-space, the gradient covering layer, and the imperfect interface. Finally, the application of the electrically open circuit, short circuit conditions, and mechanically traction-free conditions gives the frequency dispersive relation. We investigate five types of gradient profiles for the covering layer where the material parameters vary gradually from the top to the bottom, and two types of imperfect interfaces, i.e., dielectrically weakly and highly conducting but mechanically compliant interfaces. The numerical results show that the surface wave speed is sensitive to the gradient profile of the covering layer with the mechanically and dielectrically imperfect interfaces between the covering layer and the substrate.

Action of a smooth flat charged punch on the piezoelectric half-space possessing symmetry of class 6

The paper focuses on the problem of indentation of a piezoelectric half-space of symmetry of class 6 by a flat charged indenter of arbitrary shape accompanied by frictionless sliding. The research is motivated by needs in scanning probe microscopy for bone tissue and its components. The problem is solved using method of potential functions. First we solve the auxiliary problem about action of point forces normal and tangential to the boundary of the half-space and point electric charge. Solution is compared with the known one for symmetry class 6mm. Then this result is used to construct explicit solution for a flat charged punch of arbitrary shape. To the best of our knowledge, analytical solution of this kind has never been constructed for piezoelectric materials of the symmetry class 6—only for the particular case of symmetry 6mm.

Axisymmetric frictionless contact of a functionally graded piezoelectric layered half-space under a conducting punch

This paper studies the axisymmetric frictionless contact of a functionally graded piezoelectric layered half-space subjected to three typical rigid punches, i.e., flat circular punch, spherical punch and conical punch . The punch is assumed as a perfect conductor with a constant electric potential. The electro-mechanical properties of the functionally graded piezoelectric materials (FGPMs) vary exponentially along the thickness direction. By employing the Hankel integral transform technique, the problem is reduced to a set of singular integral equations of the first kind. These equations are then discretized and solved numerically to determine the unknown normal contact pressure and electric charge. The effects of the gradient index and punch geometry on the surface electro-mechanical contact behaviors are discussed in detail.

Normal point force and point electric charge in a piezoelectric transversely isotropic functionally graded half-space

Normal point force and point electric charge acting on surface of a transversely isotropic piezoelectric half-space with a functionally graded transversely isotropic piezoelectric coating are considered. Elastic moduli, piezoelectric constants and dielectric permeabilities of the coating vary with depth according to arbitrary functions. Analytical expressions for the elastic displacements and potential of the electrostatic field are derived for a fixed value of the depth coordinate. Asymptotical analysis of these formulas is derived for small and big values of a radial coordinate. An equilibrium of the half-space under the action of axisymmetric mechanical (normal and tangential) and electric loading is studied and a scheme of reducing the solutions of mixed boundary value problems to integral equations is obtained. As an illustration of the obtained solution, the PZT-5H piezoceramics with typical examples of functionally graded and homogeneous coatings are considered. The results include computations of the profiles of displacements and electric potential for different types of variation of electro-elastic properties in the coating.

Analysis of a nonlinear crack in a piezoelectric half-space via displacement discontinuity method

In this paper, we derive the exact closed-form fundamental solutions due to uniform extended displacement discontinuities over a triangular element in a piezoelectric half-space. Using the triangular elements to partition the penny-shaped crack, the triangular element fundamental solutions are verified by comparing with the existing analytical solution associated with the penny-shaped crack. The polarization saturation model is then applied to an elliptical crack in the piezoelectric half-space, and the resulting nonlinear fracture problem is solved by combing the triangular element fundamental solutions and the displacement discontinuity method. The electric yielding zone and the extended field intensity factors are obtained by an iterative approach. The effects of the applied mechanical load and electric displacement, the polarization saturation in the yielding zone, and the aspect ratio of the elliptical crack on the yielding zone size and field intensity factors are discussed through numerical examples.

Anti-plane waves near an interface between two piezoelectric half-spaces

We show the existence of certain new waves that can propagate near an interface between two half-spaces of different piezoelectric ceramics, where the interface is modeled by a membrane with the surface/interface elasticity [1]. The current configuration can be reduced into a number of well-known results as special cases, such as Love wave, Bleustein and Gulyaev wave. Together with our previous work for the imperfect interface [2], a full range of consideration of the interface affecting the anti-plane waves is now completed.

Influence of concentrated loading on opening of an interface crack between piezoelectric materials in a compressive field

A plane problem for an electrically permeable tunnel crack between two piezoelectric half-spaces under concentrated loading applied to the crack faces and remote mixed-mode loading is studied. The elastic displacement and potential jumps as well as the stresses and electrical displacement along the interface are represented via sectionally holomorphic vector functions. At the crack tip, we assume closed crack faces with frictionless contact at the crack tip. The problem is reduced to combined Dirichlet–Riemann boundary value problems that are solved analytically. From these solutions, clear analytical expressions for characteristics of the electromechanical field at the interface are derived. A transcendental equation, which determines the point of separation of the open and closed section of the crack, is found. The stress intensity factors at the crack tip and the energy release rate are calculated. The influence of the magnitude and the locations of the applied concentrated loading upon the contact zone length and the associated fracture mechanical parameters is considered.

Influences of mechanically and dielectrically imperfect interfaces on the reflection and transmission waves between two piezoelectric half spaces

The influences of mechanically and dielectrically imperfect interfaces on the reflection and transmission waves 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. Then, six kinds of imperfect interfaces are considered. These imperfect interfaces include: the mechanically compliant, dielectrically weakly conducting imperfect interface; the mechanically compliant, dielectrically highly conducting imperfect interface; the grounded metallized interface and the low-dielectric interface and their mechanical counterpart, namely, the fixed interface and the slippery interface. These imperfect interface conditions are required to be satisfied by four sets of waves, namely, the quasi-longitudinal wave (QP), the quasi-transverse wave (QSV), the shear horizontal wave (SH) which is decoupled to other waves and the electric-acoustic wave (EA). The algebraic equations resulting from the imperfect interface conditions are solved to obtain the amplitude ratio of various waves and furthermore the reflection and transmission coefficients in terms of the energy flux ratio. The numerical results are obtained for the incident QP wave, the incident QSV wave and the incident SH wave and are validated by the energy conservation principle. The effects of these imperfect interfaces are discussed based on the numerical results. The present study provides useful information for the detection of imperfect interface.

Shear-horizontal surface waves in a half-space of piezoelectric semiconductors

We study the propagation of shear-horizontal waves near the surface of a piezoelectric semiconductor half-space of crystals of class 6 mm with the presence of a biasing electric field in the propagation direction. The three-dimensional equations of linear piezoelectric semiconductors are used. A transcendental equation that determines the dispersion relation is obtained and solved numerically. Results show that the semiconduction affects the wave speed and causes wave dispersion as well as attenuation, and that the waves can be amplified by the biasing electric field.

The peculiarities of the Bleustein–Gulyaev wave propagation in structures containing conductive layer

As known, anomalous resisto-acoustic effect is a fundamental property for weakly inhomogeneous piezoactive waves (Bleustein-Gulyaev, Love, and some leaky waves). It consists in that the velocity of aforementioned waves first increases, achieves its maximum, and only then decreases with increasing conductance of a layer placed at the surface of piezoelectric half-space. In this paper we continue to study the peculiarities of the effect appearance and the influence of different electrical boundary conditions on its characteristics. Conditions have been found under which the said effect appears and, respectively, disappears. The magnitude of positive change in velocity with increasing layer conductance is demonstrated to be reduced up to zero as a layer with arbitrary conductance has been moved away from the piezoelectric surface. The positive change in velocity increases when an perfectly conductive screen moves away from the “piezoelectric half-space – conductive layer” structure. The obtained results are useful for a more deep understanding the physical basis of propagation of weakly inhomogeneous piezoactive acoustic waves (Bleustein-Gulyaev, Love, and some leaky waves).

Love-type wave propagation in a piezoelectric structure with irregularity

The present study delves to study the propagation of Love-type wave in an irregular piezoelectric layer lying over a piezoelectric half-space. The closed form expressions of dispersion relation for both the cases of electrically open and electrically short condition have been established. Wave number, irregularity, dielectric constant and piezoelectric constant of irregular piezoelectric layer and piezoelectric half-space have their substantial effect on the phase velocity of Love-type wave. Numerical computation and graphical demonstration has been carried out to observe the profound effect of these affecting parameters on the phase velocity of Love-type wave. In order to highlight some of the important peculiarities of the problem, comparative study of the phase velocity of Love-type wave propagating in an irregular piezoelectric layer lying over a piezoelectric half-space and an irregular piezoelectric layer lying over a half-space without piezoelectricity for both the cases of electrically open and electrically short condition has been made. It is remarkable to quote that the size of irregularity disfavors the phase velocity of Love-type wave for both electrically open and electrically short condition. As a special case of the problem, it is found that the obtained dispersion relation is in well agreement to the classical Love wave equation for both the cases of electrically open condition and electrically short condition.

EXPLICIT SOLUTIONS FOR AN ANISOTROPIC PIEZOELECTRIC HALF-SPACE SUBJECT TO LINEARLY-VARYING SURFACE LOADINGS ALONG x3-DIRECTION

Explicit results for a piezoelectric half-space x2≥ 0 subject to linearly-varying surface loadings along x3axis are derived. The extended Stroh formalism is employed to provide three-dimensional solutions with the generalized displacement vector u expressed as a function of (z, x2, x3). A general polynomial solution for u with order of m in x3is suggested and it provides a particularly efficient solution for half-space problem with loadings on the surface. A simple uniform surface loading is considered first to clarify the derivations. Then explicit solution in case of a linearly-varying surface loading along x3-direction is obtained. In addition, the Green's function for a piezoelectric half-space with a linearly-varying surface line loading along x3-axis is constructed.