Piezoelectric Half Research Articles

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.

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.

On some three-dimensional problems of piezoelectricity

The problem of an elliptical crack embedded in an unbounded transversely isotropic piezoelectric medium and subjected to remote normal loading is considered first. The integral equation method developed by Roy and his coworkers has been applied suitably with proper modifications to solve the problem. The method has been further applied to solve the problem of a rigid elliptical punch indenting a transversely isotropic piezoelectric half space.

Indentation responses of piezoelectric films ideally bonded to an elastic substrate

The influences of elastic substrate on the indentation force, contact radius, electric potential and electric charge responses of piezoelectric film/substrate systems are investigated by the integral transform method. The film is assumed to be ideally bonded to the substrate and the contact interaction between the indenter and the film is assumed to be frictionless, with three kinds of axisymmetric insulating and conducting indenters (i.e., punch, cone and sphere) considered. Obtained results show that when the ratio of the contact radius to the film thickness is close to zero, the influences of the elastic substrate disappear and the indentation behaviors converge to the piezoelectric half space solutions while the indentation responses approach the corresponding ones of elastic half space as the ratio gets to infinity. The transition between the piezoelectric and the elastic half space indentation solutions for the film/substrate system is quantified in terms of the film thickness and the elasticity of the substrate. Finite element analysis on an insulating sphere indentation is conducted to verify the numerical calculations and good agreement is observed. The obtained results are believed to be useful for developing experimental techniques to extract the material properties of piezoelectric film/substrate systems.

Reflection and transmission of plane waves from a fluid-porous piezoelectric solid interface

The reflection and transmission of plane waves from a fluid-porous piezoelectric solid interface is studied. The porous piezoelectric solid, having 6 mm symmetry, is supposed to be filled with viscous fluid. The expressions for amplitude ratios and energy ratios corresponding to reflected wave and transmitted waves are derived analytically. The Christoffel equation of a leaky wave propagating along the surface of a porous piezoelectric solid is derived. The effects of the angle of incidence, frequency, porosity, piezoelectric interaction, and anisotropy on the reflected and transmitted energy ratios are studied numerically for a particular model BaTiO(3). The porous piezoelectric solid half space is assumed to be loaded with water. The effects of porosity and frequency on the leaky wave velocity are also studied.

Mode III fracture problem of an arbitrarily oriented crack in an FGPM strip bonded to a homogeneous piezoelectric half plane

This paper studies the Mode III electric-elastic field of a cracked functionally graded piezoelectric strip bonded to a homogeneous piezoelectric half plane. The crack is oriented in arbitrary direction. The material properties of the strip vary along the strip thickness in exponential forms. By using the Fourier transform, the problem can be formulated to a system of singular integral equations and solved by applying the Gauss-Chebyshev integration formula. The effects come from the edge, crack orientations and the nonhomogeneous material parameter on intensity factors are discussed graphically.

Mode III fracture problem of an arbitrarily oriented crack in a FGPM strip bonded to a FGPM half plane

This paper studies the mode III electro-elastic field of a cracked functionally graded piezoelectric strip bonded to a functionally graded piezoelectric half plane. The crack is oriented in arbitrary direction. The material properties along x-axis vary in exponential form. By using the Fourier transform, the problem can be formulated into a system of singular integral equations and solved by applying the Gauss–Chebyshev integration formula. The effects come from the edge, crack orientation and the nonhomogeneous material parameters on intensity factors are discussed graphically.

Transverse surface waves on a piezoelectric material carrying a functionally graded layer of finite thickness

The propagation behavior of transverse surface waves (Love waves) in a piezoelectric half space of polarized ceramics carrying a functionally graded material layer is studied from the three-dimensional equations of linear piezoelectricity. The Wentzel–Kramers–Brillouin (WKB) asymptotic technique is adopted for the theoretical derivations of analytical solutions in the functionally graded layer. The dispersion relations of Love wave in the structure are obtained for both electrically open and shorted cases. Firstly, these solutions are used to study effect of the gradient coefficients on the dispersive relations and phase velocities of Love wave propagation. Then influence of the gradient coefficients on the electromechanical coupling factor is discussed in detail. The results reported are meaningful for the design of surface acoustic wave (SAW) devices with high performance.

Acoustic Gap Waves in Piezoelectromagnetic Materials

It is shown that the propagation of a gap wave is possible between two ceramic piezoelectric half spaces as governed by the three-dimensional equations of piezoelectromagnetism with full electromagnetic coupling.

Responses of Piezoelectric, Transversely Isotropic, Functionally Graded, and Multilayered Half Spaces to Uniform Circular Surface Loadings

Abstract: In this paper, an analytical solution is presented to study the response of piezoelectric, transversely isotropic, functionally graded, and multilayered half spaces to uniform circular surface loadings (pressure or negative electric charge). The inhomogeneous material is exponentially graded in the vertical direction and can have multiple discrete layers. The propagator matrix method and cylindrical system of vector functions are used to first derive the solution in the transformed domain. In order to find the responses in the physicaldomain, which are expressed in one-dimensional infinite integrals of the Bessel function products, we introduced and adopted an adaptive Gauss quadrature. Two piezoelectric functionally graded half-space models are analyzed numerically: One is a functionally graded PZT4 half space, and the other a multilayered functionally graded half space with two different piezoelectric materials (PZT-4 and PZT-6B). The effect of different exponential factors of the functionally graded material on the field responses is clearly demonstrated. The difference of the responses between the two surface loading cases is also discussed via the numerical examples. The results should be particularly useful in the characterization of material properties using indentation tests, and could indirectly contribute to the design and manufacturing of piezoelectric functionally graded structures.

Propagation of a mode-III interfacial conductive crack along a conductive interface between two piezoelectric materials

An analysis is presented for transient response of a mode-III interfacial crack propagating between two dissimilar piezoelectric half spaces. The dynamic fracture toughness of a piezoelectric interface is examined, which is a central issue related to the interface strength of multi-layered sensors and ferroelectric thin film devices. For the mode-III crack propagation in piezoelectric materials, an electro-acoustic surface wave, the Bleustein–Gulyaev wave, controls the crack propagation speed. It is shown that the propagation of an interfacial crack in piezoelectric media may excite an interfacial electro-acoustic surface wave, the Maerfeld–Tournois wave, which is fundamentally different from the mode-III interfacial crack propagation in purely elastic media. Moreover, it has been discovered that the existence of the Maerfeld–Tournois wave may play a significant role in determining the dynamic fracture toughness for piezoelectric composites.

Two mode III internal cracks located within two bonded functionally graded piezoelectric half planes respectively

This paper studies the mode III crack problem of two bonded functionally graded piezoelectric half planes which contain a crack respectively. These two cracks are located normal to the interface. All the material properties are assumed to vary along the direction of the crack line. A system of singular integral equations for electrically impermeable and permeable cracks is derived and solved numerically by using the Gauss–Chebyshev integration formula. The influence of the nonhomogeneous parameters and the dependence of the crack interactions on the stress and electric displacement intensity factors are investigated.

Dynamic behavior of unequal parallel permeable interface multi-cracks in a piezoelectric layer bonded to two piezoelectric materials half planes

This study is concerned with the treatment of the dynamic behavior of interacting cracks in a piezoelectric layer bonded to two dissimilar piezoelectric half planes subjected to harmonic anti-plane shear waves. The permeable electric boundary condition is considered. By use of the Fourier transform technique, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in two series of Jacobi polynomials. The electromechanical behavior of two pairs of unequal parallel cracks was determined. Numerical examples are provided to show the effects of the geometry of the cracks, the frequency of the incident waves and materials properties upon the dynamic stress intensity factors (DSIFs) and the electric displacement intensity factors.

Capillary adhesive contact between a spherical rigid punch and a piezoelectric half space

Adhesive contact between an axisymmetric punch and a piezoelectric half space in the presence of a liquid bridge formed by capillary condensation is studied. The coupling between liquid-bridge force and piezoelectric elastic deformation is taken into account. The solution is obtained by using the integral transform method with the aid of the dual integral equation technique. Combined with the thermodynamics equilibrium of capillary condensation, two competitive factors closely related to the relative humidity are shown to affect the capillary adhesive force. Two dimensionless parameters, which represent the material properties, and geometric relative relationships between liquid-bridge and contact systems, respectively, are defined to investigate the role of the capillary bridge on adhesive contact. Due to the electromechanical coupling, the applied electric load is shown to evidently affect adhesive contact. This may provide a possible means by which to weaken the influence of the capillary bridge in atomic force microscopy piezoelectric measurements.

Exact solution of a flat smooth punch on a piezoelectric half plane

This paper derives an exact solution for a flat smooth punch applied on a piezoelectric half plane. The piezoelectric solid occupies the lower half plane, and a flat rigid punch is applied on it. As the permittivity of the air (environment) is far less than that of the piezoelectric material, the electric induction of air may be neglected. The permittivity of the punch is also far less than that of the piezoelectric material and consequently the normal component of the electric displacement vanishes at the contact boundary. The exact solution is obtained by eigen-function representation and analytic continuation. The distribution of pressure under the punch has been found. The electric field along the surface of the lower half plane is extracted in a closed form.

Elastic and piezoelectric fields in a substrate AlN due to a buried quantum dot

Using the Green’s function solution in an anisotropic and linearly piezoelectric half space developed recently by the authors, this article studies the elastic and piezoelectric fields in substrate AlN due to a buried quantum dot (QD). Two different growth orientations are considered: One is the AlN (0001) growing along the (0001) axis, and the other is the AlN (1000) growing along the polar direction [i.e., a direction normal to (0001) axis]. For an InN QD, modeled as a concentrated source, with a volume va=4πR3/3 where R=3 nm, and the typical values of misfit strain, at a depth h=10 nm below the surface, the following features have been observed: (1): on the surface of substrate AlN (0001), the hydrostatic strain, piezoelectric potential, and vertical and horizontal electric fields are rotationally symmetric with respect to the z axis. However, these quantities are not rotationally symmetric on the surface of substrate AlN (1000); (2): a hydrostatic strain as large as 0.01 on the surface of the AlN (1000) and as large as 0.008 on the surface of AlN (0001) can be reached, both of which are larger than that on the surface of the substrate GaAs due to a QD with the same volume at the same depth; (3): the piezoelectric potential on the surface of substrate AlN (0001) is much larger than that on the surface of AlN (1000), i.e., 0.8 versus 0.3 V; and (4): large horizontal and vertical electric fields, in the order of 108 V/m, can be induced on the surface of AlN, about two orders of magnitude larger than that on the surface of the substrate GaAs due to a QD with the same volume at the same depth.

A moving conducting crack at the interface of two dissimilar piezoelectric materials

The problem of a Yoffe-type conducting crack moving with a constant velocity at the interface of two dissimilar piezoelectric half planes is investigated by employing complex variable method. Solutions for the complex potentials are derived. Explicit expressions for the field components on the interface are presented based on the obtained complex potentials. It is observed that the nature of the field singularities near the crack tip is intimately dependent on the crack moving velocity. In the extremely low speed regime, the singularities are δ=−1/2±i ε 1; in the low speed regime, the singularities are δ=−1±i ε 2; in the intermediate speed regime, the singularities are δ=−1/2± k; in the high speed regime, the singularities are δ=−1±i ε 3; in the extremely high speed regime, the singularities are δ=−1/2±i ε 4. ε i (i=1–4) and k are also explicitly given. A Yoffe-type moving conducting crack in a homogeneous piezoelectric material is treated as a special case. The numerical results demonstrate that the moving velocity V will exert a significant influence on the value of the singularities, and on the field component distributions along the interface.

Elastic and piezoelectric fields in substrates GaAs (001) and GaAs (111) due to a buried quantum dot

In this article we present a rigorous study on the elastic and piezoelectric fields in substrates GaAs (001) and GaAs (111) due to a buried quantum dot (QD) using an efficient and accurate continuum mechanics model. It is based on a Green’s function solution in anisotropic and linearly piezoelectric half space combined with the generalized Betti reciprocal theorem. To address the effect of material anisotropy, two other substrates, Iso (001) and Iso (111), are also examined and they are assumed to be elastically isotropic. For a point QD with hydrostatic misfit strain γ*=0.07 in volume va=4πa3/3 where a=3 nm, and at depth h=10 nm below the surface, we have observed the following features. (1) The simplified elastically isotropic model should, in general, not be used for predicting elastic and piezoelectric fields in the semiconductor GaAs. (2) The magnitude of the QD-induced piezoelectric potential on the surface of GaAs (111) or GaAs (001) is comparable to, or even larger than, the direct potential. (3) Large horizontal and vertical electric fields, on the order of 106 V/m, can be induced on the surface of GaAs (001) and GaAs (111). (4) The elastic field induced on the surface of GaAs (001) has rotational symmetry of order C4 (i.e., the elastic field remains the same after rotation of 2π/4 around the [001] axis), while the corresponding piezoelectric field has rotational symmetry of order C2. On the other hand, both the elastic and piezoelectric fields on the surface of GaAs (111) have rotational symmetry of C3 around the [111] axis. (5) The magnitude of the elastic and piezoelectric quantities on the surface of GaAs (111) is, in general, larger than that of the corresponding quantities on the surface of GaAs (001). (6) Under different electric surface conditions (insulating or conducting), the surface piezoelectric fields induced are quite different.