Dispersion relations of elastic waves in three-dimensional cubical piezoelectric phononic crystal with initial stresses and mechanically and dielectrically imperfect interfaces

A three-dimensional cubical piezoelectric phononic crystal is theoretically studied in this paper, formed by pre-stressed piezoelectric rectangular blocks and imperfect interfaces. Firstly, transfer matrices of pre-stressed piezoelectric rectangular blocks and imperfect interfaces are obtained based on the structural characteristics. The Bloch waves consist of three groups of sub-coupled elastic waves corresponding with three orthogonal periodical directions in the three-dimensional periodical structure. Furthermore, based on the transfer matrices of typical single cell and the Bloch theorem, it is established that the theoretical model of above-mentioned three-dimensional cubical piezoelectric phononic crystal to obtain the dispersion relations of Bloch waves. Finally, the influences of non-dimensional geometrical parameters (structural and modal parameters) and physical parameters (initial stress and mechanically and dielectrically imperfect interface parameters) on the dispersion relations are discussed based on the graphically numerical results. Numerical calculation results show that the existences of initial stresses and mechanically and dielectrically imperfect interfaces equivalently lead to the alterations of structural flexibilities or rigidities. The theoretical model and numerical discussions will provide a direct guidance of multi-material additive manufacturing for pre-stressed and inhomogeneous periodic structures with partial and global dispersion properties.     

Effects of functionally graded interlayers on dispersion relations of shear horizontal waves in layered piezoelectric/piezomagnetic cylinders

The effects of functionally graded interlayers on dispersion relations of shear horizontal waves in layered piezoelectric/piezomagnetic cylinders are studied. First, the basic physical quantities of elastic waves in piezoelectric cylinder are derived by assuming that the SH waves propagate along the circumferential direction steadily. Then the transfer matrices of the functional graded interlayer and outer piezomagnetic cylinder are obtained by solving the state transfer equations with spatial-varying coefficients. Furthermore, making use of the electro-magnetic surface conditions of the outer cylinder, the dispersion relations for the shear horizontal waves in layered piezoelectric/piezomagnetic cylinders are obtained and the numerical results are shown graphically. Seven kinds of functionally graded interlayers and four kinds of electro-magnetic surface conditions are considered. It is found that the functionally graded interlayers have evident influences on the dispersion relations of shear horizontal waves in layered piezoelectric/piezomagnetic cylinders. The high order modes are more sensitive to the gradient interlayers while the low order modes are more sensitive to the electro-magnetic surface conditions.     

二维各向异性压电介质机电耦合场的基本解

本文研究各向异性压电介质的机电耦台问题．应用平面波分解法和留数定理，首次得到了线力和线电荷作用下一般二维各向异性压电介质机电耦合场的基本解．本文的解适用于平面问题、反平面问题以及平面和反平面相互耦合问题．作为特例，文中给出了横观各向同性压电介质的基本解．     

Transient response of a semi-infinite piezoelectric layer with a surface permeable crack

The transient response of a semi-infinite transversely isotropic piezoelectric layer containing a surface crack is analyzed for the case where anti-plane mechanical and in-plane electric impacts are suddenly exerted at the layer end. The integral transform techniques are used to reduce the associated mixed initial boundary value problem to a singular integral equation of the first kind, which can be solved numerically via the Lobatto–Chebyshev collocation technique. Dynamic field intensity factors are determined by employing a numerical inversion of the Laplace transform. The dynamic stress intensity factors are presented graphically and the effects of the material properties and geometric parameters are examined. Received: June 30, 2003     

不同功能梯度压电压磁层状介质中共线界面裂纹的动态性能分挝

对不同功能梯度压电压磁层状介质中,共线界面裂纹对简谐应力波作用下的动态问题,进行了分析.经Fourier变换,使问题的求解转换为求解以裂纹面上位移间断为未知量的三重对偶积分方程,三重对偶方程可以采用Schmidt方法来求解,进而分析了功能梯度参数、入射波频率和层状介质厚度对应力、电位移和磁通量强度因子的影响.     

Displacement potentials for functionally graded piezoelectric solids

Two new displacement potential functions are introduced for the general solution of a three-dimensional piezoelasticity problem for functionally graded transversely isotropic piezoelectric solids. The material properties vary continuously along the axis of symmetry of the medium. The four coupled equilibrium equations in terms of displacements and electric potential are reduced to two decoupled sixth- and second-order linear partial differential equations for the potential functions. The obtained results are verified with two limiting cases: (i) a functionally graded transversely isotropic medium, and (ii) a homogeneous transversely isotropic piezoelectric solid. The simplified relations corresponding to the special case of similar variation of material properties are also given. Furthermore, the special cases of axisymmetric problems, exponentially graded piezoelectric media and transversely isotropic piezoelectric media with power law variation are discussed in detail.     

An asymptotic approach to the problem of the distant propagation of waves

An asymptotic approach is used in this paper to examine solutions of the problem of distant propagation of Love waves in a medium consisting of an inhomogeneous layer lying on a homogeneous half-space. The dispersion properties of these waves and their attenuation are estimated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 125–135, 1986.     

Poling angle effect on two mode-III semi-permeable collinear cracks in a piezoelectric strip: Strip-saturation model

A mathematical strip-saturation model is proposed to analyze the influence of changing poling direction on two hairline electrically semi-permeable collinear cracks embedded in a piezoelectric strip. The remote boundaries of the piezoelectric strip are subjected to anti-plane mechanical and in-plane electrical loads. Fourier series method is applied to transform the model into triple series equations which are then reduced to singular integral equations using the integral equation technique. The closed-form analytical expressions are derived for saturation zones length, crack-sliding displacement (CSD), crack opening potential drop (COPD), intensity factors (IFs), and energy release rate (ERR). Influence of the poling angle on CSD, COPD, and ERR are shown graphically for various cracked piezoelectric ceramics. In addition, the numerical study is also presented for analyzing the effect of inter-crack distance and prescribed electrical and mechanical loadings on the fracture parameters. All graphical results are analyzed and compared.     

On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular/annular plates

In recent years many articles concerned with the mechanics of functionally graded plates have been published. The variation in material properties through the thickness of the plate introduces a coupling between in-plane and transverse displacements, the coupling is important in the vibration of functionally graded plates (FGPs), but none have produced an exact closed-form solution for the in-plane as well as transverse vibrations of smart circular/annular FGPs. Therefore, this paper develops an exact closed-form solution for the free vibration of piezoelectric coupled thick circular/annular FGPs subjected to different boundary conditions on the basis of the Mindlin’s first-order shear deformation theory. Through the comparison of present results with those available, the accuracy of the present method was verified. The effects of coupling between in-plane and transverse displacements on the frequency parameters are proved to be significant. It is concluded that the developed model can describe vibrational behavior of smart FGM plates more realistic. Due to the inherent features of the present solution, all findings will be a useful benchmark for evaluating other analytical and numerical methods developed by researchers in the future.     

An interface crack moving between magnetoelectroelastic and functionally graded elastic layers

A constant crack moving along the interface of magnetoelectroelastic and functionally graded elastic layers under anti-plane shear and in-plane electric and magnetic loading is investigated by the integral transform method. Fourier transforms are applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are expressed in terms of Fredholm integral equations of the second kind. The singular stress, electric displacement and magnetic induction near the crack tip are obtained asymptotically and the corresponding field intensity factors are defined. Numerical results show that the stress intensity factors are influenced by the crack moving velocity, the material properties, the functionally graded parameter and the geometric size ratios. The propagation of the moving crack may bring about crack kinking, depending on the crack moving velocity and the material properties across the interface.     

压电压磁复合材料中一对平行裂纹对弹性波的散射

利用Schmidt方法对压电压磁复合材料中一对平行对称裂纹对反平面简谐波的散射问题进行了分析,借助富里叶变换得到了以裂纹面上的间断位移为未知变量的对偶积分方程．在求解对偶积分方程的过程中,裂纹面上的间断位移被展开成雅可比多项式的形式,最终获得了应力强度因子、电位移强度因子、磁通量强度因子三者之间的关系．结果表明,压电压磁复合材料中平行裂纹动态反平面断裂问题的应力奇异性与一般弹性材料中的动态反平面断裂问题的应力奇异性相同,同时讨论了裂纹间的屏蔽效应．     

3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory

This paper addresses a 3D elasticity analytical solution for static deformation of a simply-supported rectangular micro/nanoplate made of both homogeneous and functionally graded (FG) material within the framework of modified couple stress theory. The plate is assumed to be resting on a Winkler–Pasternak elastic foundation, and its modulus of elasticity is assumed to vary exponentially along thickness. By expanding displacement components in double Fourier series along in-plane coordinates and imposing relevant boundary conditions, the boundary value problem (BVP) of plate system, including its governing partial differential equations (PDEs) of equilibrium are reduced to BVP consisting only ordinary ones (ODEs). Parametric studies are conducted among displacement and stress components developed in the plate and FG material gradient index, length scale parameter, and foundation stiffnesses. From the numerical results, it is concluded that the out-of-plane shear stresses are not necessarily zero at the top and bottom surfaces of plate. The results of this investigation may serve as a benchmark to verify further bending analyses of either homogeneous or FG micro/nanoplates on elastic foundation.     

Effects of initial stress on guided waves in orthotropic functionally graded plates

Because of the limitation of the manufacturing technology, initial stress in functionally graded materials (FGM) and structures is inevitable. Based on the theory of “Mechanics of Incremental Deformations”, the guided wave propagation in FGM plates under gravity, homogeneous initial stress in the thickness direction and inhomogeneous initial stress in the wave propagation direction is investigated. The Legendre polynomial series method is used to solve the coupled wave equations with variable coefficients. The convergence of the polynomial series method is discussed through the numerical examples. The effects of the initial stress on the Lamb-like wave and on the SH wave are investigated respectively and the numerical results show they are quite distinct. The effect of the gravity on the wave propagation can be ignored. The effects of the initial stress in the thickness direction are very different from those of the initial stress in the wave propagation direction, both on the dispersion curves and on the displacement and stress distributions.     

有限高狭长压电体中共线双半无限裂纹问题的解析解

通过引入合适的保角变换,利用复变函数法,分析了部分裂纹面上受反平面剪应力和面内电载荷共同作用下有限高狭长压电体中含共线双半无限裂纹问题,导出了电不可通边界条件下两个裂纹尖端场强度因子和机械应变能释放率的解析解．当不考虑电场作用时,所得解可退化到经典弹性材料的情况．而当两裂纹尖端的距离趋于无穷大时,也可退化为狭长压电体中半无限裂纹问题的解．最后,通过数值算例,讨论了受载长度、狭长体高度、机电载荷对机械应变能释放率的影响规律以及两个裂纹之间的相互作用．结果表明,两裂纹尖端的距离越短,材料越容易破坏;且机电载荷对左尖端裂纹的扩展影响更为显著．     

X-ray tomographic determination of the chemical composition and structure of an inhomogeneous medium

The chemical composition of an inhomogeneous body consisting of several homogeneous parts is determined by x-ray tomography. At the first stage, an indicator of inhomogeneities is used to determine the internal structure of the body. Next, under certain additional assumptions about the properties of the parts, a method is proposed for partial or complete determination of the chemical composition of each part as based on the results of the previous stage. Mathematically, the problem is reduced to solving the radiative transfer equation and systems of linear algebraic equations. Numerical experiments are performed via computer simulation. The numerical results are illustrated by graphs and tomograms.     

Dirichlet-Neumann conditions and the orthogonality of three-dimensional guided waves in layered solids

Orthogonality relations for homogeneous waves in layered plates are obtained, and they are generalized to the case of a contact with fluid layers. For layers of infinite thickness, it is shown that the homogeneous waves of the discrete spectrum are orthogonal to each other and to the waves of the continuous spectrum. For finite-size sources, exact formulas are derived for the coefficients multiplying the modes. Based on the orthogonality relations, a nonlocal radiation principle is proposed such that the infinite domain in the numerical solution of diffraction problems for layered plates can be replaced by a virtual cylinder.     

On dispersion relations of waves in multilayered magneto-electro-elastic plates

The dispersion behavior of waves in a layered magneto-electro-elastic plate is studied. Formulations are given for open-circuit surface and short-circuit surface, respectively. Legendre orthogonal polynomial series approach is employed to solve the coupling magneto-electro-elastic controlling equations. The obtained dispersion curves for the open-circuit surface are compared with the published data to confirm the validity of the method. Through numerical examples, the convergence of the method is discussed. The differences of the two boundaries are illustrated. The influences of the piezoelectricity and piezomagnetism are illustrated and the influential factors of the piezoelectric and piezomagnetic effect are discussed. Finally, the influences of the volume fraction and stacking sequence on the piezoelectric effect and piezomagnetic effect are discussed.     

夹层压电材料中垂直于界面的共线双裂纹动力学问题分析

采用Schmidt方法分析了在简谐反平面剪切波作用下,两个半空间夹层压电材料中的共线裂纹的动力学行为．压电材料层内裂纹垂直于界面,电边界条件假设为可导通．通过Fourier变换,使问题的求解转换为两对三重积分对偶方程．通过数值计算,给出了裂纹的几何尺寸、压电材料常数、入射波频率等对于应力强度因子的影响．结果表明,在不同的入射波频率范围,动力场将阻碍或促使压电材料内裂纹的扩展．与不可导通电边界条件相比,导通裂纹表面的电位移强度因子比不可导通裂纹的电位移强度因子要小许多．     

Dispersion Equations for Rayleigh Waves in a Piezoelectric Periodically Layered Structure

Dispersion equations are proposed for acoustoelectric Rayleigh waves in a periodically layered piezoelectric half space with various types of boundary conditions. The properties of the medium are specified by the determining relations for the 6mm crystallographic class. These equations are obtained using the mathematical formalism of periodic hamiltonian systems. This approach makes it possible to include the anisotropy and the piezoelectric interaction of the mechanical and electric fields and is valid for stratified media with arbitrary variations in the properties along the periodicity axis. Numerical results are presented for alternating layers of CdS and ZnO. The influence of the piezoelectric effect and type of boundary conditions on the dispersion spectra of surface waves is examined.     

Propagation of shear waves in piezoelectric “superlattice-substrate” structures

The results of a computer simulation of the dispersion relations for the propagation of shear waves in piezoelectric “superlattice-substrate” structures are analyzed. The superlattice consists of a finite number of layers and is made up of materials with 6mm symmetry. The dispersion relations are obtained using a formulation for periodic hamiltonian systems. This approach makes it possible to account for the anisotropy, the piezoelectric interaction of the mechanical and electric fields, and an arbitrary number of layers in the superlattice. Numerical results are presented for CdS-ZnO layers. Selective spatial localization of acoustic modes is demonstrated for different spectral regions. The effects of the ordering of the superlattice layers, of their number, and of the boundary conditions on the dispersion spectra and on the form of the shear wave motion are examined. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, and Institute of Metal Physics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 148–156, 1999.