Semi-infinite Piezoelectric Medium Research Articles

Theoretical analysis of surface waves in piezoelectric medium with periodic shunting circuits

The investigations of surface waves in the piezoelectric medium bring out great possibility in designing smart surface acoustic wave (SAW) devices. It is important to study the dispersion properties and manipulation mechanism of surface waves in the semi-infinite piezoelectric medium connected with periodic arrangement of shunting circuits. In this study, the extended Stroh formalism is developed to theoretically analyze the dispersion relations of surface waves under different external circuits. The band structures of both the Rayleigh wave and the Bleustein-Gulyaev (BG) wave can be determined and manipulated with proper electrical boundary conditions. Furthermore, the electromechanical coupling effects on the band structures of surface waves are discussed to figure out the manipulation mechanism of adjusting electric circuit. The results indicate that the proposed method can explain the propagation behaviors of surface waves under the periodic electrical boundary conditions, and can provide an important theoretical guidance for designing novel SAW devices and exploring extensive applications in practice.

Generalized Thermopiezoelectricity with Memory-Dependent Derivative and Transient Thermoelectromechanical Responses Analysis

In this paper, a novel generalized thermoelastopiezoelectric model is established by introducing memory-dependent derivative, which might be superior to fractional ones: the form is unique, while the fractional-order theories have various expressions with different authors; it is more intuitionistic for reflecting the memory effect; the physical meanings of the related memory-dependent differential equations are more clear, which are determined by the essence of their definitions; the time-delay and kernel function can be chosen freely based on the necessity of practical applications. The newly constructed model is applied to the transient shock analysis for piezoelectric medium under heating loads. Laplace transformation techniques are employed to solve the governing equations. In numerical implementation, the problem of a semi-infinite piezoelectric medium is considered under the two different cases. The transient responses, that is, temperature, displacement, stress, and electric potential, are illustrated graphically. The parametric studies are performed to analyze the effects of time-delay and kernel function on the transient thermoelastopiezoelectric responses. This work may provide a new approach to explore the transient responses’ behavior for piezoelectric materials serving in nonisothermal environment.

Dynamic anti-plane behavior for a semi-infinite piezoelectric medium with a locally debonded cylindrical inclusion near the surface

The problem of dynamically debonded cylindrical inclusion near the interface of semi-infinite piezoelectric materials was theoretically investigated. The effects of different geometric and physical parameters on the dynamic stress intensity factor (DSIF) of the crack tip are discussed. The theoretical expressions for the crack (debonding) DSIF were obtained using methods that included Green’s function, the complex variable function, and multipolar coordinates, and the numerical results showed that the dynamic characteristics of the debonded structure were more obvious under conditions of low frequency and large piezoelectric characteristic parameters. In addition, the period of the DSIF at the crack tip became shorter as the incident wave number increased. There are, therefore, important theoretical and engineering considerations for the dynamic analysis of piezoelectric materials with debonded cylindrical inclusion.

Screw dislocations in piezoelectric laminates with four or more phases

We present an analytical solution to the problem of a screw dislocation in a four-phase piezoelectric laminate composed of two piezoelectric layers of equal thickness sandwiched between two semi-infinite piezoelectric media. A new version of the complex variable formulation is proposed such that the 2 × 2 real symmetric matrix appearing in the formulation becomes dimensionless. Using analytic continuation, the original boundary value problem is reduced to the identification of a single 2D analytic vector function which is completely determined following rigorous solution of the resulting linear recurrence relations in matrix form. An explicit expression for the image force acting on the piezoelectric screw dislocation is obtained once the single 2 × 2 real matrix function is identified. We also discuss the solution for a screw dislocation in an N -phase piezoelectric laminate composed of N − 2 piezoelectric layers of equal thickness sandwiched between two semi-infinite piezoelectric media.

Screw dislocation near a piezoelectric oblique edge crack

The interaction between a screw dislocation and an oblique edge crack in a semi-infinite piezoelectric medium under remote anti-plane mechanical and in-plane electrical loadings is studied in this article. The screw dislocation suffers a finite discontinuity in the displacement and electric potential across the slip plane. The dislocation also has a line force and a line charge along its core. In the framework of linear piezoelectricity theory, the analytical solutions are derived from the complex variable and Schwarz–Christoffel transformation. Furthermore, the electric field, stress field, generalized stress intensity factor, image force and crack extension force are formulated exactly. Numerical calculations are then given graphically for studying the effects of crack length, crack angle and load position on these electric mechanical parameters. Besides, the dislocation solutions proposed here can be served as Green’s functions for the analyses of corresponding piezoelectric cracking or distributed loading problems.

The reflection phenomena of quasi-vertical transverse waves in piezoelectric medium under initial stresses

The propagation of plane vertical transverse waves at an interface of a semi-infinite piezoelectric elastic medium under the influence of the initial stresses is discussed. The free surface of the piezoelectric elastic medium is considered to be adjacent to vacuum. We assumed that the piezoelectric material is anisotropic of the type of a transversely isotropic crystals (hexagonal crystal structure, class 6 mm). For an incident of vertical transverse plane wave, four types (two for the displacement and two for the electric potential) of reflected plane waves, called quasi-longitudinal (qP) and quasi-shear vertical (qSV) waves are shown to be exist. The relations governing the reflection coefficients of these reflected waves for various boundary conditions (mixed-free-fixed) are derived. It has been shown analytically that reflected coefficients of (qP) and (qSV) waves depend upon the angle of incidence, the parameters of electric potential, the material constants of the medium as well as the initial stresses presented in the medium. The numerical computations of reflection coefficients for different values of initial stresses have been carried out by computer for aluminum nitride (AlN) as an example and the results are given in the form of graphs. Finally, particular cases are considered in the absence of the initial stresses and the electric potential. Some of earlier studies have been compared to the special cases and shown good agreement with them.

Fractional Order Generalized Thermo-Piezoelectric Semi-Infinite Medium with Temperature-Dependent Properties Subjected to a Ramp-Type Heating

The generalized thermoelasticity theory that, based on a fractional order model, is used to solve a one-dimensional boundary value problem of a semi-infinite piezoelectric medium. The resulting formulation is applied to a half-space subjected to ramp-type heating and traction free. The generalized thermo-piezoelectricity model in an isotropic elastic medium with temperature-dependent mechanical properties is established. The modulus of elasticity is taken as a linear function of the reference temperature. The Laplace transform technique is used to obtain the general solution for any set of boundary conditions. The inverse Laplace transforms are numerically computed using the Fourier expansion techniques. The effects of fractional order and the ramping of heating parameters are studied and comparison with different theories of thermoelasticity are considered. The results are also compared to results obtained in the case of a temperature-independent modulus of elasticity.

Dynamic Stress Intensity Factor for a Radial Crack on a Circular Cavity in a Piezoelectric Medium

The problem of dynamic stress intensity factor is investigated theoretically in present paper for a radial crack on a circular cavity in an infinite piezoelectric medium, which is subjected to time-harmonic incident anti-plane shearing. First, a pair of electromechanically coupled Green’s functions are constructed which indicate the basic solutions for a semi-infinite piezoelectric medium with a semi-circular cavity. Second, based on the crack-division technique and conjunction technique, integral equations for the unknown stresses’ solution on the conjunction surface is established, which are related to the dynamic stress intensity factor at the crack tip. Third, the analytical expression on dynamic stress intensity factor at the crack tip is obtained. At last, some calculating cases are plotted to show how the frequencies of incident wave, the piezoelectric characteristic parameters of the material and the geometry of the crack and the circular cavity influence upon the dynamic stress intensity factors. While some of the calculating results are compared with the same situation about a straight crack and with static solutions.

Dynamic Anti-Plane Behaviors of a Semi-Infinite Piezoelectric Medium with a Circular Cavity and a Crack near the Surface

Dynamic interaction is investigated theoretically between a circular cavity and a crack near the surface in a semi-infinite piezoelectric medium subjected to time-harmonic incident anti-plane shearing in this paper. The formulations are based on the method of complex variable and Green’s function. Dynamic stress concentration factors at the edge of the circular cavity and dynamic stress intensity factors at the crack tip are obtained by solving boundary value problems with the method of orthogonal function expansion. The calculating results are plotted to show how the frequencies and the orientation of incident wave, piezoelectric characteristic parameters of the material and the structural geometries influence upon the dynamic stress concentration factor (DSCF) and dynamic stress intensity factor (DSIF).

Numerical analysis of the properties of slit electroacoustic waves

The properties of slit electroacoustic waves that propagate in a system of two semi-infinite piezoelectric media separated by a vacuum gap, in a system consisting of a thin piezoelectric plate and a semi-infinite piezoelectric medium separated by a gap, and in a system consisting of two thin piezoelectric plates separated by a vacuum gap are studied. The process of transformation of slit electroacoustic waves to generalized surface acoustic waves or to Lamb waves is considered.

Thermal Shock Strength of a Semi-infinite Piezoelectric Medium

This paper considers the mechanical problem of a semi-infinite piezoelectric medium under sudden thermal load. The medium contains an electrically conducting crack perpendicular to its surface. The transient stresses and electric fields in an uncracked medium are calculated first. Then, these stresses and electric fields are used as the crack surface traction and electric field loads with opposite signs to formulate the mixed boundary value problem. Numerical results for the stress and electric field intensity factors are calculated as a function of normalized time and crack size. Crack propagation behavior is discussed. The parameters that control the transient thermal stress and electric fields are also identified. The maximum thermal shock strength that the material can sustain without catastrophic failure is established according to two distinct criteria: (i) maximum local tensile stress equals the tensile strength of the medium, and (ii) maximum stress intensity factor for the preexisting representative crack equals the fracture toughness of the medium. The results show that the influence of the piezoelectric effects on the thermal stress intensity factor is insignificant.

Analytical Solution for Shear Horizontal Wave Propagation in Piezoelectric Coupled Media by Interdigital Transducer

An analytical solution for the shear horizontal wave propagation excited by interdigital transducer in a piezoelectric coupled semi-infinite medium is developed. This solution is an extension of earlier work on wave propagation in a piezoelectric coupled plate with finitely long interdigital transducer by fully taking account of piezoelectric effects in analysis. In the current analysis, the mathematical model for a semi-infinite metal substrate bonded by a layer of interdigital transducer with infinite length is first derived. The theoretical solutions are obtained in terms of elliptic integration of the first kind and of the standard integral representation for Legendre polynomial. The essential hypothesis for the derivation of the analysis is investigated. Based on the solution for infinitely long interdigital transducer, an analytical solution for the wave propagation in this semi-infinite piezoelectric medium excited by a finitely long interdigital transducer is obtained through Fourier transform. This theoretical research can be applied to health monitoring of structures by interdigital transducer. It could also be used as a framework for the design of interdigital transducer in wave excitation of smart structures.

Surface fracture of a semi-infinite piezoelectric medium under transient thermal loading (poling axis parallel to the edge of the medium)

In [Mech. Mater. 34 (2002) 135–144], surface fracture of a semi-infinite piezoelectric medium under transient thermal load was considered for the case of the poling axis perpendicular to the edge of the medium. Investigated in this paper is the thermal fracture for another important case. Namely, the poling direction is parallel to the edge of the medium. In this case, both stresses and electric displacements exist ahead of the crack tip. The transient temperature and the stress distributions in an uncracked medium are calculated. The stress and the electric displacement for an uncracked medium are used as the crack surface traction with opposite sign to formulate the mixed boundary value problem. Results for stress intensity factors are computed as a function of the normalized time and the crack size. The crack propagation behavior is also discussed.

Surface thermal shock fracture of a semi-infinite piezoelectric medium (poling axis parallel to the crack plane)

Considered in this paper is a semi-infinite piezoelectric medium containing an edge crack perpendicular to the surface of the medium. The medium is subjected to a sudden cooling on its surface. The transient temperature and the stress distributions in an uncracked strip are calculated. This stress for uncracked medium is used as the crack surface traction with opposite sign to formulate the mixed boundary value problem. A singular integral equation of Cauchy-type is derived and solved numerically. The numerical results for the stress intensity factor are computed as a function of the normalized time and the crack size. The temperature and the thermal stress distributions for the uncracked problem are also included.

On existence of one-partial bulk waves in semi-infinite piezoelectric media

The conditions for the existence of one-partial bulk waves satisfying the boundary conditions on a mechanically free surface of a semi-infinite piezoelectric medium have been analyzed. In purely elastic media, similar wave solutions are known to exist along the propagation directions m which form lines on the sphere m ⋅ m = 1, passing through all the available degeneracy points (acoustic axes). It is shown that in a triclinic piezoelectric half-space with a metallized surface, one-partial bulk waves may exist solely along isolated propagation orientations, whereas for a nonmetallized surface, such waves can exist only if an additional condition for the material constants of the medium is fulfilled. It is also shown that the one-partial bulk solution may not exist along an arbitrary acoustic axis in a piezoelectric. These conclusions are valid in the general case of unrestricted anisotropy, i.e., they do not take into account the material symmetry. In addition, the orientations providing the propagation of one-partial bulk waves because of the existing symmetry are specified for piezoelectric media of various symmetry classes.

Shear interfacial waves in piezoelectrics

An analysis was made of the SH-interfacial wave propagation on an interface between semi-infinite piezoelectric media. In particular, a sufficient condition for the existence of the SH-wave on the metallized rigid contact was derived. It has been shown that on the metallized rigid contact between two completely identical piezocrystals the SH-localized wave always exists, with the exception of at most two orientations of the interface. An investigation was performed of specific features of the sound reflection—transmission at incident angles corresponding to the resonant excitation of the leaky wave originating from the “supersonic” interfacial wave due to misorientation of crystals that constitute the bicrystal. Also, it has been established that in 6 mm crystals an interfacial “supersonic” wave may remain purely localized even when the direction of propagation is not perpendicular to the axis 6 or when the interface is not parallel to this axis.

State vector solutions for nonaxisymmetric problem of multilayered half space piezoelectric medium

A state space formulation is established for the nonaxisymmetric space problem of transversely isotropic piezoelectric media in a system of cylindrical coordinate by introducing the state vector. Using the Hankel transform and the Fourier series, the state vector equations are transformed into ordinary differential equations. By the use of the matrix methods, the analytical solutions of a single piezoelectric layer are presented in the form of the product of initial state variables and transfer matrix. The applications of state vector solutions are discussed. An analytical solution for a semiinfinite piezoelectric medium subjected to the vertical point forceP z, horizontal point forceP x along x-direction and point electric charge Q at the origin of the surface is presented. According to the continuity conditions at the interfaces, the general solution formulation forN-layered transversely isotropic piezoelectric media is given.

Disturbances in a Semi-Infinite Piezoelectric Medium Coated with a Perfectly Conducting Thin Film under the Influence of Time-Dependent Flow of Heat for a Finite Interval

An attempt was made to investigate analytically the mechanical behaviour of a semi-infinite piezoelectric medium coated with a perfectly conducting thin film under the influence of time-dependent flow of heat at the finite end. The problem involves the interaction of three fields, viz. electrical, mechanical and thermal, and the solution was obtained with the aid of operational calculus. The resultant displacement consists of a time-dependent, as well as a time-independent part and the variation of displacements exhibits an arc of a parabola. PACS numbers: 77.70.+a

Harmonic oscillations of an electroelastic semi-infinite medium, caused by a periodic action in space

A two-dimensional boundary value problem of the harmonic oscillations of a semi-infinite piezoelectric medium with one flat boundary on which a normal displacement and an electrical field potential are given periodically, is considered. This problem occurs in the design of a number of surface acoustic wave devices /1/. Such devices consist of a piezoelectric crystal of rectangular planform and cross-section on one of whose faces a periodic system of rectangular electrodes is superimposed. The presence of the periodically arranged electrodes on the boundary exerts an influence on the surface acoustic waves by two means: 1) electrical shorting of the surface, and 2) mechanical action on the oscillating medium because of electrode inertia. The contribution of the mechanical action here grows as the operating frequencies of the device increase. The boundary value problem reduces to a system of periodic convolution equations. The properties of the kernels of the integral equations are established. A theorem is presented that enables one to transfer to the solution of systems of algebraic equations. A solution is constructed for the wave fields at any point of the medium. An example is considered for calculating the wave fields on the boundary of the medium.

Some exact solutions for the propagation of transient electroacoustic waves. II: Plane interface between two piezoelectric media

Electroacoustic waves are coupled horizontal-shear (SH) and electromagnetic disturbances that propagate in certain types of piezoelectric media. At the interface between two semi-infinite piezoelectric media, a new type of electroacoustic surface wave, called a Maerfeld-Tournois wave, can exist that has no purely elastic wave counterpart. This article obtains exact transient solutions for the Maerfeld-Tournois and body waves generated by a dipole source on such an interface. These solutions are obtained for both conducting and nonconducting interfaces using a modification of the Lamb-Cagniard-Pekeris technique. In the quasistatic approximation, explicit results for all the waves present are given and the separated body and surface waves at the interface are discussed.