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．     

Interior and boundary difference equations for hyperbolic differential equations

Interior and boundary difference equations are derived for several hyperbolic partial differential equations by means of an integral method. The method is applied to a simple transport equation, to waves in a compressible, isentropic fluid, and to surface waves in shallow water. Boundary conditions treated are (a) a perfectly reflecting boundary, (b) an open boundary with outgoing waves and a specified incoming wave, and (c) a partially reflecting boundary. For open boundaries, the major assumption for the algorithms to be valid is that outgoing waves can be defined, an assumption equivalent to the most general statement of Sommerfeld's radiation condition. The difference equations obtained are conservative, second-order accurate, two time-level, explicit, and stable (for one-dimensional, time-dependent problems) for cΔt/Δx ? 1 where c is the wave speed, Δt is the temporal grid size, and Δx is the spatial grid size. Numerical calculations demonstrate the excellent accuracy of the procedure.     

Global bifurcation of travelling waves in discrete nonlinear Schrödinger equations

The discrete nonlinear Schrödinger equations of n sites are studied with periodic boundary conditions. These equations have n branches of standing waves that bifurcate from zero. Travelling waves appear as a symmetry-breaking from the standing waves for different amplitudes. The bifurcation is proved using the global Rabinowitz alternative in subspaces of symmetric functions. Applications to the Schrödinger and Saturable lattices are presented.     

Complex guided waves in functionally graded piezoelectric cylindrical structures with sectorial cross-section

For the purpose of the design and optimization of piezoelectric transducers, the modified double orthogonal polynomial series method is proposed to investigate guided waves in functionally graded piezoelectric(FGP) cylindrical structures with sectorial cross-section. The real, imaginary and complex solutions are obtained simultaneously without iterative process. The real solutions represent propagative waves; the imaginary and complex solutions are evanescent waves. The boundary conditions are incorporated into the constitutive equations by virtue of the Heaviside function. Subsequently, the amplitudes are expanded into the double orthogonal polynomial series, and the motion equations are converted into a matrix eigenvalue problem about complex wavenumber. Numerical comparison with available reference result confirms the validity of the present method. Dispersion curves and the Poynting vector distributions are illustrated. The influences of angular measure, radius-thickness ratio and graded index on dispersion curves are analyzed. Results show that there exist some evanescent guided wave modes that have higher velocities than that of the propagative wave modes and simultaneously have low attenuation at high frequencies. These results can be utilized to improve the performance of transducers.     

Surface effects on the buckling behaviors of piezoelectric cylindrical nanoshells using nonlocal continuum model

The focus of this paper is on the analytical buckling solutions of piezoelectric cylindrical nanoshells under the combined compressive loads and external voltages. To capture the small-scale characteristics of the nanostructures, the constitutive equations with the coupled nonlocal and surface effects are adopted within the framework of Reddy's higher-order shell theory. The governing equations involving the displacements and induced piezoelectric field are solved by employing the separation of variables. The derived accurate solutions conclude that bucking critical stresses should go down rapidly while the nonlocal effects reach a certain level. With the enhancing surface effects, the stability of piezoelectric cylindrical nanoshells can be improved significantly. Meanwhile, the induced electric field also plays an important role in elevating the buckling critical stresses. For the nanoshells with remarkable nonlocal effects, boundary conditions, shell length and radius have little influence on the buckling solutions. The detailed effects of the boundary conditions, geometric parameters, material properties and applied voltages are discussed.     

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.     

主被动阻尼层合板结构的自由振动和阻尼特性分析

给出了含主被动阻尼非对称复合材料层合板结构的振动微分方程;得到了在压电材料和高粘弹材料作主被动阻尼层情况下,简支层合板结构自由振动的自然频率和损失因子的解析解;分析了正逆向压电效应对自然频率和损失因子的影响     

Theoretical and Experimental Investigations of Piezoelectrically Excited Travelling Waves in Cylindrical Tubes

Flügge's equations of motion for cylindrical shells are used to analyze the generation of travelling waves in a tube of an ultrasonic travelling wave transport system. The vibration-shape of a semi-infinite acrylic tube with structural damping excited by a piezoelectric ring at the end is calculated and the influence of the driving frequency is shown. Starting with an exponential ansatz the frequency dependent wavenumbers can be determined. Three frequency bands with different combinations of imaginary, real and complex wavenumbers can be found. Supposing a semi-infinite tube with given boundary conditions at the end of excitation the vibrations of the tube can be calculated depending on the frequency. Simulation agrees well with experimental results. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Far-Field Equations in Linear Elasticity for Disconnected Rigid Bodies and Cavities

In this paper the far-field equations in linear elasticity for scattering from disjoint rigid bodies and cavities are considered. The direct scattering problem is formulated in differential and integral form. The boundary integral equations are constructed using a combination of single- and double-layer potentials. Using a Fredholm type theory it is proved that these boundary integral equations are uniquely solvable. Assuming that the incident field is produced by a superposition of plane incident waves in all directions of propagation and polarization it is established that the scattered field is also expressed as the superposition of the corresponding scattered fields. A pair of integral equations of the first kind which hold independently of the boundary conditions are constructed for the far-field region. The properties of the Herglotz functions are used to derive solvability conditions for the far-field equations. It is also proved that the far-field operators, in terms of which we can express the far-field equations, are injective and have dense range. An analytical example for spheres illuminates the theoretical results.     

Low Mach Number Limit of Viscous Polytropic Flows: Formal Asymptotics in the Periodic Case

This article is devoted to the low Mach number limit of weak solutions to the compressible Navier–Stokes equations for polytropic fluids with periodic boundary conditions and ill‐prepared data. We derive formally the equation satisfied by the mean value of the velocity and the equations governing the dynamics of the nonlinear acoustic waves in dimension d= 2 or 3.     

Free vibration of a functionally graded annular sector plate integrated with piezoelectric layers

Based on the first order shear deformation theory, free vibration behavior of functionally graded (FG) annular sector plates integrated with piezoelectric layers is investigated. The distribution of electric potential along the thickness direction of piezoelectric layers which is assumed to be a combination of linear and sinusoidal functions, satisfies both open and closed circuit electrical boundary conditions. Through a reformulation of governing equations and harmonic motion assumption, a novel decoupling method is suggested to transform the six second order coupled partial differential equations of motion into two eighth order and fourth order equations. A Fourier series method is then employed to present analytical solutions for free vibration of smart FG annular sector plates with simply supported radial edges and arbitrarily supported circular edges. The results, which can be used as a benchmark and suitable for design purposes, are verified with those reported in the literature. Finally, by presenting extensive ranges of frequencies, the effects of geometric parameters, power law index, FG and piezoelectric materials, electrical and mechanical boundary conditions as well as the piezoelectric layer thickness on vibration response of smart annular sector plates are discussed in detail.     

简单波的解析解

Riemann于1860年给出了简单波的一般解,可是欲将其中所含的一个任意函数的具体形式按边界条件或初始条件求出来,却难以实现,因而对具体问题就难以深入探讨.本文按拟线性偏微分方程的几何理论,十分方便地给出了由边界条件或初始条件所确定的简单波解析解,并用这些解讨论了有关流动的性质,且得到了若干新颖的结果     

On Love-type waves in a finitely deformed magnetoelastic layered half-space

In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney?CRivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein?CGulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper.     

Elastodynamical Scattering by N Parallel Half-Planes in IR3

Linear boundary value problems of elasticity describe the propagation of time- harmonic waves outside of N parallel half-plane shaped cracks in the Euclidian 3-space. Equivalent systems involving 6N Wiener-Hopf equations are obtained for first, second and third kind conditions simultaneously. To find explicit solutions, complex-valued matrix functions with nonrational entries, are to be factorized in a generalized manner. This is done for two double-knife screen crack problems in Part II. Problems for waves in acoustics, hydro-and electrodynamics with an analogous geometry for rigid walls, or perfectly conducting metallic sheets, are contained in the problems formulated above: In Part II, for pure Dirichlet-, or Neumann conditions, the corresponding (reduced) Wiener-Hopf operator is seen to be invertible by an operator Neumann series for all distances (≠ 0) between the N half-planes Σm.     

Initial boundary value problems for nonlinear dispersive wave equations

In this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half-line and on a finite interval subject to homogeneous Dirichlet boundary conditions. We first prove local well-posedness of the rod equation and of the b-equation for general initial data. Furthermore, we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span. In the case of finite time singularities we are able to describe the precise blow-up scenario of breaking waves. Our approach is based on sharp extension results for functions on the half-line or on a finite interval and several symmetry preserving properties of the equations under discussion.     

Open boundary conditions for hyperbolic equations

A previously developed general procedure for deriving accurate difference equations to describe conditions at open boundaries for hyperbolic equations is extended and further illustrated by means of several examples of practical importance. Problems include those with both incoming and outgoing waves at the boundary, the use of locally cylindrical and spherical wave approximations at each point of the boundary, and nonlinear wave propagation. Reflected waves in all cases are minimal and less than 10^?2 of the incident wave.     

Dissipative Boundary Conditions for Finite-Difference Problems in Elasticity Theory

Dissipative differential-difference boundary conditions for elasticity equations are derived and investigated. These conditions completely eliminate the longitudinal and transverse waves reflected from the boundary in case of normal incidence and substantially reduce the reflected waves in a wide range of other incidence angles. The dissipative boundary conditions reduce computer resource requirements for simulation of seismic wave propagation in unbounded regions. Grid analogues of the dissipative boundary conditions are constructed for a two-layer finite-difference scheme. Comparative results are reported for the solution of two-dimensional seismic problems using these boundary conditions for various incidence angles. __________ Translated from Prikladnaya Matematika i Informatika, No. 18, pp. 83–92, 2004.     

An explicit exact analytical approach for free vibration of circular/annular functionally graded plates bonded to piezoelectric actuator/sensor layers based on Reddy’s plate theory

In the present study, a novel exact closed-form procedure based on the third order shear deformation plate theory is developed to analyze in-plane and out-of-plane frequency responses of circular/annular functionally graded material (FGM) plates embedded in piezoelectric layers for both close/open circuit electrical boundary conditions. Introducing a new analytical method, five governing partial deferential equations of motion beside Maxwell electrostatic equation are solved via an exact closed-form method. The high accuracy and reliability of the present approach is confirmed by comparing some of the present data with their counterparts reported in the literature. Finally, the effect of material properties, power law index and boundary conditions on the free vibration of the smart FGM plate are studied and discussed in detail.     

Electro-mechanical shear buckling of piezoelectric nanoplate using modified couple stress theory based on simplified first order shear deformation theory

This paper studies the electro-mechanical shear buckling analysis of piezoelectric nanoplate using modified couple stress theory with various boundary conditions.In order to be taken electric effects into account, an external electric voltage is applied on the piezoelectric nanoplate. The simplified first order shear deformation theory (S-FSDT) has been employed and the governing differential equations have been obtained using Hamilton's principle and nonlinear strains of Von-Karman. The modified couple stress theory has been applied to considering small scale effects. An analytical approach was developing to obtain exact results with various boundary conditions. After all, results have been presented by change in some parameters, such as; aspect ratio, effect of various boundary conditions, electric voltage and length scale parameter influences. At the end, results showed that the effect of external electric voltage on the critical shear load occurring on the piezoelectric nanoplate is insignificant.     

Buckling and free vibration analysis of high speed rotating carbon nanotube reinforced cylindrical piezoelectric shell

In this paper, buckling and free vibration behavior of a piezoelectric rotating cylindrical carbon nanotube-reinforced (CNTRC) shell is investigated. Both cases of uniform distribution (UD) and FG distribution patterns of reinforcements are studied. The accuracy of the presented model is verified with previous studies and also with those obtained by Navier analytical method. The novelty of this study is investigating the effects of critical voltage and CNT reinforcement as well as satisfying various boundary conditions implemented on the piezoelectric rotating cylindrical CNTRC shell. The governing equations and boundary conditions have been developed using Hamilton's principle and are solved with the aid of Navier and generalized differential quadrature (GDQ) methods. In this research, the buckling phenomena in the piezoelectric rotating cylindrical CNTRC shell occur as the natural frequency is equal to zero. The results show that, various types of CNT reinforcement, length to radius ratio, external voltage, angular velocity, initial hoop tension and boundary conditions play important roles on critical voltage and natural frequency of piezoelectric rotating cylindrical CNTRC shell.