Love waves in a layered functionally graded piezoelectric structure under initial stress

Abstract

To study the effect of initial stress on the propagation behavior of Love waves in a layered functionally graded piezoelectric structure, a mathematical model is established. The piezoelectric layer is taken as exponentially graded material where as half-space is taken as simply elastic substratum. The coupled electromechanical field equations are solved analytically to obtain the mechanical displacements and electrical potential functions for the piezoelectric layer and elastic substrate. The dispersion relations are obtained for electrically open and short cases. The higher mode Love wave propagation has been considered. For numerical interpretation of the results, four sets of piezoelectric layer and elastic substrate have been taken into consideration. Graphical representation reveals about the effect of initial stress and the effect of inhomogeneity parameter on the phase velocity against wave number for electrically open and electrically short cases, respectively.     

Analysis of transient responses in a laminated piezoelectric cylindrical shell

A hybrid numerical method is proposed for analysis of transient responses in a multilayered piezoelectric cylindrical shell.In the present method,the associated equations of the displacement field and the electro-potential field are developed using an analytical-numerical method.The piezoelectric cylindrical shell is discretized into layered annular elements along the wall thickness direction.The governing equations are determined by Hamilton's Principle considering the coupling between the elastic and elec...     

The refined theory of transversely isotropic piezoelectric rectangular beams

The problem of deducing one-dimensional theory from two-dimensional theory for a transversely isotropic piezoelectric rectangular beam is investigated. Based on the piezoelasticity theory, the refined theory of piezoelectric beams is derived by using the general solution of transversely isotropic piezoelasticity and Lur’e method without ad hoc assumptions. Based on the refined theory of piezoelectric beams, the exact equations for the beams without transverse surface loadings are derived, which consist of two governing differential equations: the fourth-order equation and the transcendental equation. The approximate equations for the beams under transverse loadings are derived directly from the refined beam theory. As a special case, the governing differential equations for transversely isotropic elastic beams are obtained from the corresponding equations of piezoelectric beams. To illustrate the application of the beam theory developed, a uniformly loaded and simply supported piezoelectric beam is examined.     

FREE VIBRATIONS OF SIMPLY SUPPORTED AND MULTILAYERED MAGNETO-ELECTRO-ELASTIC PLATES

Analytical solutions are derived for free vibrations of three-dimensional, linear anisotropic, magneto-electro-elastic, and multilayered rectangular plates under simply supported edge conditions. For any homogeneous layer, we construct the general solution in terms of a simple formalism that resembles the Stroh formalism, from which any physical quantities can be solved for given boundary conditions. In particular, the dispersion equation that characterizes the relationship between the natural frequency and wavenumber can be obtained in a simple form. For multilayered plates, we derive the dispersion relation in terms of the propagator matrices. The present solution includes all previous solutions, such as piezoelectric, piezomagnetic, and purely elastic solutions as special cases, and can serve as benchmarks to various thick plate theories and numerical methods used for the modelling of layered composite structures. Typical natural frequencies and mode shapes are presented for sandwich piezoelectric/piezomagnetic plates. It is shown clearly that some of the modes are purely elastic while others are fully coupled with piezoelectric/piezomagnetic quantities, with the latter depending strongly upon the material property and stacking sequence. These frequency and mode shape features could be of particular interest to the analysis and design of various “smart” sensors/actuators constructed from magneto-electro-elastic composite laminates.     

Electromechanical response of 2-2 layered piezoelectric composites: A micromechanical model based on the asymptotic homogenization method

A micromechanical model based on the asymptotic homogenization technique has been developed to predict the complete elastic, dielectric and piezoelectric properties of a general 2-2 layered piezoelectric composite where the constituent phases are elastically anisotropic and piezoelectrically active. Two classes of layered piezoelectric composites (i.e. longitudinally and transversely layered) are considered in two widely different ceramic- and polymer-based systems and their effective properties are obtained in the limits of both large-volume (i.e. bulk) and small-volume (i.e. thin-film) systems. It is demonstrated that: (i) in the bulk, ceramic–ceramic layered composite system, the elastic, piezoelectric, and dielectric properties of the composites vary linearly with volume fraction of the second phase, while in the bulk ceramic–polymer layered composite system, the corresponding properties vary non-linearly with volume fraction of the second phase; (ii) in the prismatic (thin-film) layered piezoelectric composite system, the non-vanishing, effective elastic, piezoelectric and dielectric properties vary linearly with the volume fraction of the second phase for both the longitudinally and transversely layered composite structures in the ceramic–ceramic and the ceramic–polymer composite systems; (iii) the ceramic–polymer piezoelectric layered composites that incorporate a low density polymeric phase with lower acoustic impedance generally exhibit enhanced piezoelectric coupling constants and lowered acoustic impedance; (iv) the longitudinally layered composites exhibit higher piezoelectric coupling constants and lower acoustic impedance compared to that of the transversely layered composites; and (v) the best combination of properties for applications such as hydrophones (i.e. the highest piezoelectric coupling constants and the lowest acoustic impedance) is obtained in the ceramic–polymer, longitudinally layered, thin-film, piezoelectric composites.     

Metamaterials with dielectric and metallic inclusions in the cubic lattice

The method of integral equations, which is based on the Green function of periodically arranged sources, is used for analyzing periodic metamaterials (photonic crystals) in the form of the simplest metallic and dielectric inclusions into a rectangular cubic lattice in a dielectric medium (matrix). Dielectric inclusions in the form of parallelepipeds and cubes are considered, as well as similar metallic inclusions (perfectly conducting metal rods) and 1D nanosized structures with metallic layers. Metal inclusions are investigated in the case of ideal conduction and in the case of penetration of a field into the metal, which is simulated as an electron plasma. The results are applicable from microwave of optical frequencies.     

Receiving sensitivity and transmitting voltage response of a fluid loaded spherical piezoelectric transducer with an elastic coating

A method is presented to determine the response of a spherical acoustic transducer that consists of a fluid-filled piezoelectric sphere with an elastic coating embedded in infinite fluid to electrical and plane-wave acoustic excitations. The exact spherically symmetric, linear, differential, governing equations are used for the interior and exterior fluids, and elastic and piezoelectric materials. Under acoustic excitation and open circuit boundary condition, the equation governing the piezoelectric sphere is homogeneous and the solution is expressed in terms of Bessel functions. Under electrical excitation, the equation governing the piezoelectric sphere is inhomogeneous and the complementary solution is expressed in terms of Bessel functions and the particular integral is expressed in terms of a power series. Numerical results are presented to illustrate the effect of dimensions of the piezoelectric sphere, fluid loading, elastic coating and internal material losses on the open-circuit receiving sensitivity and transmitting voltage response of the transducer.     

Three-dimensional analytical solution for a transversely isotropic functionally graded piezoelectric circular plate subject to a uniform electric potential difference

This paper studies the problem of a functionally graded piezoelectric circular plate subjected to a uniform electric potential difference between the upper and lower surfaces. By assuming the generalized displacements in appropriate forms, five differential equations governing the generalized displacement functions are derived from the equilibrium equations. These displacement functions are then obtained in an explicit form, which still involve four undetermined integral constants, through a step-by-step integration which properly incorporates the boundary conditions at the upper and lower surfaces. The boundary conditions at the cylindrical surface are then used to determine the integral constants. Hence, three-dimensional analytical solutions for electrically loaded functionally graded piezoelectric circular plates with free or simply-supported edge are completely determined. These solutions can account for an arbitrary material variation along the thickness, and thus can be readily degenerated into those for a homogenous plate. A numerical example is finally given to show the validity of the analysis, and the effect of material inhomogeneity on the elastic and electric fields is discussed. Supported by the National Natural Science Foundation of China (Grant Nos. 10472102 and 10432030) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20060335107)     

Dispersion of waves and characteristic wave surfaces in functionally graded piezoelectric plates

An inhomogeneous layer element method is presented to analyze the dispersion of waves and characteristic wave surfaces in plates of functionally graded piezoelectric material (FGPM). In this method, the FGPM plate is divided into a number of layered elements. The elemental elastic and electric properties are assumed as linear functions of the thickness to adopt the variety of the material property of FGPM. The Hamilton principle is applied to determine the governing equations. The phase velocity surface, phase slowness surface, phase wave surface, group velocity surface, group slowness surface, and group wave surface for FGPM plate are formulated using Rayleigh quotient and the orthogonality condition of the eigenvectors. These six surfaces are then used to illustrate the characteristics of waves in FGPM plates. Numerical examples are presented using the present formulations to analyze dispersions and characteristics of waves in FGPM plates.     

Wave propagation in layered piezoelectric rectangular bar: An extended orthogonal polynomial approach

Wave propagation in multilayered piezoelectric structures has received much attention in past forty years. But the research objects of previous research works are only for semi-infinite structures and one-dimensional structures, i.e., structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper proposes an extension of the orthogonal polynomial series approach to solve the wave propagation problem in a two-dimensional (2-D) piezoelectric structure, namely, a multilayered piezoelectric bar with a rectangular cross-section. Through numerical comparison with the available reference results for a purely elastic multilayered rectangular bar, the validity of the extended polynomial series approach is illustrated. The dispersion curves and electric potential distributions of various multilayered piezoelectric rectangular bars are calculated to reveal their wave propagation characteristics.     

Higher-dimensional chaotic dynamics of a composite laminated piezoelectric rectangular plate

The analysis on the chaotic dynamics of a six-dimensional nonlinear system which represents the averaged equation of a composite laminated piezoelectric rectangular plate is given for the first time. The theory of normal form and the energy-phase method are combined to investigate the higher-dimensional chaotic dynamics of the composite laminated piezoelectric rectangular plate. Firstly, the theory of normal form is used to reduce the six-dimensional averaged equation to the simpler normal form. Then, the energy-phase method is extended to analyze the global bifurcations and chaotic dynamics of a six-dimensional nonlinear system. The analysis results indicate that there exist the homoclinic bifurcation and Shilnikov type multi-pulse chaos for the composite laminated piezoelectric rectangular plate. Finally, numerical simulations are also used to investigate the nonlinear dynamic characteristics of the composite laminated piezoelectric rectangular plate. The results of numerical simulations also demonstrate that there exist the chaotic motions and the multi-pulse jumping orbits of the composite laminated piezoelectric rectangular plate.     

Size-dependent dispersion characteristics in piezoelectric nanoplates with surface effects

In this paper, surface effects on the dispersion characteristics of elastic waves propagating in an infinite piezoelectric nanoplate are investigated by using the surface piezoelectricity model. Based on the surface piezoelectric constitutive theory, the presence of surface stresses and surface electric displacements exerting on the boundary conditions of the piezoelectric nanoplate is taken into account in the modified mechanical and electric equilibrium relations. The partial wave technique is employed to obtain the general solutions of governing equations, and the dispersion relations with surface effects are expressed in an explicit closed form. The impacts of surface piezoelectricity, residual surface stress and plate thickness on the propagation properties of elastic waves are analyzed in detail. Numerical results show that the dispersion behaviors in piezoelectric nanoplates are size-dependent, and there exists a critical plate thickness above which the surface effects may vanish.     

A note on pure bending of a piezoelectric rectangular plate

The problem of pure bending of a piezoelectric rectangular plate is considered in this paper by using the electromagnetic equations of Maxwell and equations of elasticity along with the constitutive equations of a piezoelectric quartz.The author extends his sincere gratitude to Dr. D. K. Sinha of Jadavpur University for his kind help and guidance in the preparation of this paper.     

Piezoacoustic wave spectra using improved surface impedance matrix: application to high impedance-contrast layered plates

Starting from the general modal solutions for a homogeneous layer of arbitrary material and crystalline symmetry, a matrix formalism is developed to establish the semianalytical expressions of the surface impedance matrices (SIM) for a single piezoelectric layer. By applying the electrical boundary conditions, the layer impedance matrix is reduced to a unified elastic form whether the material is piezoelectric or not. The characteristic equation for the dispersion curves is derived in both forms of a three-dimensional acoustic SIM and of an electrical scalar function. The same approach is extended to multilayered structures such as a piezoelectric layer sandwiched in between two metallic electrodes, a Bragg coupler, and a semi-infinite substrate as well. The effectiveness of the approach is numerically demonstrated by its ability to determine the full spectra of guided modes, even at extremely high frequencies, in layered plates comprising up to four layers and three materials. Negative slope in f-k curve for some modes, asymptotic behavior at short wavelength regime, as well as wave confinement phenomena made evident by the numerical results are analyzed and interpreted in terms of the surface acoustic waves and of the interfacial waves in connection with the bulk waves in massive materials.     

General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect

By introducing four potential functions, the governing equations of plane problems in 1D orthorhombic quasicrystals with piezoelectric effect are composed of four second-order partial differential equations, in which the quasi-harmonic functions are the essential unknowns. The general solution of these equations is further established, and all expressions are expressed in terms of the potential functions. As an application of the general solution, the closed-form solutions are obtained for wedge problems or half-plane problems of 1D orthorhombic piezoelectric quasicrystals.     

Low-frequency scattering by rectangular dielectric particles

The field scattered by a homogeneous isotropic dielectric particle illuminated by a low-frequency plane electromagnetic wave is expressed in terms of a single polarizability tensor which is a function of only the geometry of the particle and a material parameter τ representing either the relative permittivity or permeability of the dielectric. The mathematical formulation is specialized to the case of a rectangular parallelepiped and numerical techniques are developed for computing the tensor elements. Specific data are presented for the tensor elements of rectangular parallelepipeds having square cross sections and are compared to the results obtained for spheroids and right circular cylinders of similar dimensions.     

Numerical investigation of air free convection in a room with heat source

Mathematical modelling of three-dimensional steady free convective incompressible viscous gas flows in the rooms with a heat source is carried out within the framework of the Navier — Stokes equations with the effective viscosity determined on the basis of the κ-ɛ turbulence model. The investigation is carried out for the case of model rooms and heat sources having the form of rectangular parallelepipeds with square bases. The influence of the heat source power and the sizes of the room base on local and averaged values of the air velocity and temperature in the rooms is analysed. The flow pattern in the room is shown to have a torus-like shape. It is found that the variation of sizes of the room base rather than the capacity variation of the heat source is of determining importance for the gas motion character in a closed volume.     

The directivity pattern of an interferometer for measuring thermal acoustic radiation

The directivity patterns of a pair of piezoelectric transducers for measuring the spatial correlation function of sound pressures produced by sources of thermal acoustic radiation in the megahertz frequency range are calculated. Sources in the form of a heated plane or strip are considered. The signal detection by two circular or rectangular piezoelectric transducers and by focusing transducers is studied. It is demonstrated that, for measuring the correlation function, the piezoelectric transducers must partially overlap. To determine the directivity pattern with a strong dependence on the distance between the heated object and the pair of piezoelectric transducers, focusing piezoelectric transducers should be used. The results obtained offer possibilities for a noninvasive measurement of the absorption coefficient of a medium and also for the realization of the previously proposed [20] passive acoustic thermotomograph, which does not use a priori information on the absorption coefficient of the medium.     

基于平面问题的位移压力混合配点法

引入压力变量,将弹性力学控制方程表达为位移和压力的耦合偏微分方程组,采用重心插值近似未知量,利用重心插值微分矩阵得到平面问题控制方程的矩阵形式离散表达式.采用重心插值离散位移和应力边界条件,采用附加法施加边界条件,得到求解平面弹性问题的过约束线性代数方程组,采用最小二乘法求解过约束方程组,得到平面问题位移数值解.数值算例验证了所提方法的有效性和计算精度.