PMN-PT夹层弹性半空间结构中SH波的传播Propagation of SH waves in sandwich structures consisting of PMN-PT single crystal layer and elastic half-spaces

分析了弹性上下半空间和PMN‐PT单晶层组成的夹层结构中SH波的传播性质,PMN‐PT单晶沿[011]c方向极化,宏观上呈mm2对称,且晶体沿角度θ方向切割。基于正交各向异性压电材料和各向同性弹性材料的基本方程,得到了夹层结构中SH波传播时行列式形式的频散方程。通过对数值算例进行分析可以看出,PMN‐PT单晶的切割角度和弹性材料属性对结构中的相速度有很大影响,因此波的某些传播性能可以通过材料的设计以及晶体切割的方向来实现,这些结论为声表面波器件的开发和应用提供了理论依据。     

Reflection and refraction of plane quasi-longitudinal waves at an interface of two piezoelectric media under initial stresses

This paper is devoted to study a problem of reflection and refraction of quasi-longitudinal waves under initial stresses at an interface of two anisotropic piezoelectric media with different properties. One of the two media is aluminum nitride, which is considered the down piezoelectric medium and the above medium is chosen as PZT-5H ceramics. The two piezoelectric media welded are assumed to be anisotropic of a type of a transversely isotropic crystals (hexagonal crystal structure, class 6 mm). The equations of motion and constitutive relations for the piezoelectric media have been written. Suitable boundary conditions are used to obtain the reflection and refraction coefficients. For an incidence of quasi-longitudinal plane waves, four independent-type amplitude ratios of elastic displacement components for plane waves, called quasi-longitudinal (qP) and quasi-shear vertical (qSV) waves, are shown to exist. Also, it is observed that there exist four dependent amplitude ratios of electric potential, which are proportional to the previous four types. Finally, it is found that the coefficients of reflection and refraction are functions of angle of incidence, elastic constants, piezoelectric potential parameters and the initial stresses. Numerical computations and the results obtained are depicted graphically. In the end, a particular case has been reduced from the present study. This investigation is considered important because the initial stresses in such practical problems are inevitable and may result in frequency shift, a change in the velocity of surface waves and controlling the selectivity of a filter compensation of the devices.     

An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

Non-linear free surface flows and an application of the Orlanski boundary condition

The focus of this paper is the analysis of spatially two-dimensional non-linear free surface problems. The critical aspects of the problem concern the treatment of the non-linear free surface, the body boundary condition for large motions and the imposition of suitable radiation conditions. To address such complexities, time domain simulation was chosen as the method of analysis. With the use of a finite domain for simulation, a major concern is with the radiation condition to be applied at the open or truncation boundary. For the two-dimensional problem at hand, no theoretical radiation conditions are known to exist. An extension of the Orlanski open boundary condition, based on phase velocity determination at the free surface, is proposed. Three categories of problems were analysed using numerical simulation-namely, freely moving steep waves, waves over a submerged body and forced body motion. Simulation results have been compared with linear theory and experiments.     

The role of divergent terms in the Frank energy of nematic liquid crystals

The effect of divergent terms in the Frank orientation energy of nematic liquid crystals on the equilibrium state of the director field is studied. Such terms have no effect on the equations of motion or on the equilibrium of the medium under consideration; however, they should be taken into account in the derivation of boundary conditions. It is shown that, in the case of boundary perturbations or in the case of polar orientation angle perturbations, the divergent terms can be considered as a surface energy for the azimuth angle (this energy is similar to the Rapini-Papoular energy). In addition, these terms may cause a deviation of the director in the plane parallel to the boundary. The equilibrium problem for a nematic liquid crystal is considered as an example in the case of small periodic boundary perturbations.     

Twelve shear surface waves guided by clamped/free boundaries in magneto-electro-elastic materials

It is shown that surface waves with 12 different velocities in the cases of different magneto-electrical boundary conditions can be guided by the interface of two identical magneto-electro-elastic half-spaces. The plane boundary of one of the half-spaces is clamped while the plane boundary of the other one is free of stresses. The 12 velocities of propagation of these surface waves are obtained is explicit forms. It is shown that the number of different surface wave velocities decreases from 12 to 2 if the magneto-electro-elastic material is changed to a piezoelectric material.     

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.     

Propagation of plane waves in thermoelastic cubic crystal material with two relaxation times

A problem concerned with the reflection and refraction of thermoelastic plane waves at an imperfect interface between two generalized thermally conducting cubic crystal solid half-spaces of different elastic and thermal properties with two relaxation times has been investigated. The generalized thermoelastic theory with two relaxation times developed by Green and Lindsay has been used to study the problem. The expressions for the reflection and refraction coefficients which are the ratios of the amplitudes of reflected and refracted waves to the amplitude of incident waves are obtained for an imperfect boundary and deduced for normal stiffness, transverse stiffness, thermal contact conductance, slip and welded boundaries. Amplitude ratios of different reflected and refracted waves for different boundaries with angle of emergence have been compared graphically for different incident waves. It is observed that the amplitude ratios of reflected and refracted waves are affected by the stiffness and thermal properties of the media.     

Some Exactly Solvable Problems of the Radiation of Three–Dimensional Periodic Internal Waves

An exact solution of the problem of the generation of three–dimensional periodic internal waves in an exponentially stratified, viscous fluid is constructed in a linear approximation. The wave source is an arbitrary part of the surface of a vertical circular cylinder which moves in radial, azimuthal, and vertical directions. Solutions satisfying exact boundary conditions, describe both the beam of outgoing waves and wave boundary layers of two types: internal boundary layers, whose thickness depends on the buoyancy frequency and the geometry of the problem, and viscous boundary layers, which, as in a homogeneous fluid, are determined by kinematic viscosity and frequency. Asymptotic solutions are derived in explicit form for cylinders of large, intermediate, and small dimensions relative to the natural scales of the problem.     

DYNAMIC BEHAVIOR OF TWO COLLINEAR PERMEABLE CRACKS IN A PIEZOELECTRIC LAYER BONDED TO TWO HALF SPACES

IntroductionIn the fracture mechanics studies for piezoelectric materials,differently electricboundary conditions at the crack surfaces have been proposed by many researchers.Forexample,for the sake of analytical simplification,the assumption that the cra…     

Elastic impact on finite Timoshenko beam

In this paper the analytical solutions of the impact of a particle on Timoshenko beams with four kinds of different boundary conditions are obtained according to Navier's idea, which is further developed. The initial values of the impact forces are exactly determined by the momentum conservation law. The propagation of the longitudinal and transverse waves along the beam, especially, the effects of boundary conditions on the characteristics of the reflected waves, are investigated in detail. Some results are compared with those by MSC/NASTRAN.     

Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

Nonstationary motion of a shell on the surface of a heavy fluid

A three-dimensional nonstationary problem of vibrations of a flexible shell moving on the surface of an ideal heavy fluid. The forces due to surface tension are ignored. The problem is formulated in the space of the acceleration potential. The potential of the pulsating source is found by solving the Euler equation and the continuity equation taking into account the free-surface conditions (linear theory of small waves) and the conditions at infinity. The density distribution function of the dipole layer is determined from the boundary conditions on the surface of the shell. Formulas for determining the shape of gravity waves on the fluid surface and the natural frequencies of vibrations of the shell are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 66–75, July–August, 2009.     

Analysis of transient wave scattering from an inclusion with an unilateral smoothly contact interface by BEM

Summary A 2D time-domain Boundary Element Method (BEM) is applied to solve the problem of transient scattering of plane waves by an inclusion with a unilateral smooth contact interface. The incident wave is assumed strong enough so that localized separations take place along the interface. The present problem is indeed a nonlinear boundary value problem since the mixed boundary conditions involve unknown intervals (separation and contact regions). In order to determine the unknown intervals, an iterative technique is developed. As an example, we consider the scattering of plane waves by the cross section of a circular cylinder embedded in an infinite solid. Numerical results for the near field solutions are presented. The distortion of the response waves and the variation of the interface states are discussed. The financial support by the China National Natural Science Foundation under Grant No. 19872001 and No. 59878004 is gratefully acknowledged. The second author is also grateful to the support of the National Science Fund for Distinguished Young Scholars under Grant No. 10025211.     

DYNAMIC INTERACTION OF PLANE WAVES WITH A UNILATERALLY FRICTIONALLY CONSTRAINED INCLUSION-TIME DOMAIN BOUNDARY ELEMENT ANALYSIS

A 2D time domain boundary element method (BEM) is developed to solve the transient scattering of plane waves by a unilaterally frictionally constrained inclusion. Coulomb friction is assumed along the contact interface. The incident wave is assumed strong enough so that localized slip and separation take place along the interface. The present problem is in effect a nonlinear boundary value problem since the mixed boundary conditions involve unknown intervals (slip, separation and stick regions). In order to determine the unknown intervals, an iterative technique is developed. As an example, we consider the scattering of a circular cylinder embeddedin an infinite solid.     

Adiabatic shear band localization of inelastic single crystals in symmetric double-slip process

Summary The main objective of the present paper is the development of a viscoplastic regularization procedure valid for an adiabatic dynamic process for multi-slips of single crystals. The next objective is to focus attention on the investigation of instability criteria, and particularly on shear band localization conditions.To achieve this aim, an analysis of acceleration waves is given, and advantage is taken of the notion of the instantaneous adiabatic acoustic tensor. If zero is an eigenvalue of the acoustic tensor, then the associated discontinuity does not propagate, and one speaks of a stationary discontinuity. This situation is referred to as the strain localization condition, and corresponds to a loss of hyperbolicity of the dynamical equations. It has been proved that for an, adiabatic process of rate-dependent (elastic-viscoplastic) crystal, the wave speed of discontinuity surface always remains real and different from zero. It means that for this case the initial-value problem is well-posed. However, for an adiabatic process of rate-independent(elastic-plastic) crystal, the wave speed of discontinuity surface can be equal zero. Then the necessary condition for a localized plastic deformation along the shear band to be formed is as follows: the determinant of the instantaneous adiabatic acoustic tensor is equal to zero. This condition for localization is equivalent to that obtained by using the standard bifurcation method. Based on this idea, the conditions for adiabatic shear band localization of plastic deformation have been investigated for single crystals. Particular attention has been focused on the discussion of the influence of thermal expansion, thermal plastic, softening and spatial covariance effects on shear band localization criteria for a planar model of an f.c.c. crystal undergoing symmetric primary-conjugate double slip. The results obtained have been compared with available experimental observations.Finally, it is noteworthy that the viscoplasticity regularization procedure can be used in the developing of an unconditionally stable numerical integration algorithm for simulation of adiabatic inelastic flow processes in ductile single crystals, cf. [21].The paper has been prepared within research programme sponsored by the Committee of Scientific Research under Grant 3 P404 031 07.     

均匀流中双色波传播特性的模拟研究

针对双色波浪与均匀流相互作用问题,采用时域高阶边界元方法建立自由水面满足完全非线性边界条件的数学模型。求解中采用混合欧拉-拉格朗日方法追踪流体瞬时水面,运用四阶龙格库塔方法更新下一时间步的波面和速度势。通过与已发表试验结果对比,验证了本模型的准确性。通过数值计算研究了水流参数对各组成波及衍生的高阶波幅值、波浪和水流间能量交换的影响规律。     

Dynamic frictional indentation of an elastic half-plane by a rigid punch

Dynamic rigid indentation of a linearly elastic half-plane in the presence of Coulomb friction is studied in this paper. A rigid punch, which is either wedge- or parabolic-shaped, is rapidly driven into the deformable body so that stress waves are generated. The contact region is assumed to extend at a constant sub-Rayleigh speed (this situation can be achieved by conveniently specifying the kinetic and geometric characteristics of indentor), whereas, due to symmetry, friction acts in opposing directions on opposite sides of the indentor. As the present exact analysis shows, this sign reversal of the tangential traction along the half-plane surface creates an extra stress-singularity at the changeover point of the boundary conditions (due to symmetry, this point here coincides with the point where the indentor apex makes contact with the half-plane surface). The study exploits the problem's self-similarity by utilizing homogeneous-function techniques previously used by L.M. Brock, along with the Riemann-Hilbert problem analysis. Representative numerical results are given for the wedge indentation case.     

单侧摩擦接触约束夹杂物与平面波的动力相互作用--时域边界元分析

应用时域边界元法研究了瞬态平面波对单侧摩擦约束夹杂物的散射问题，假设界面摩擦遵守库仑定理，当入射波足够强时界面会出现局部分离和滑移，由于边界上的区域(分离区、滑移区和粘着区)是未知的，所以该问题实际上是个复杂的边界非线性问题，为了确定未知区域，该文发展了一种有效的迭代技术，作为算例，计算了一个无限域中圆柱埋置夹杂物对瞬态平面波的散射问题。