Dyadic Green's functions for cylindrical waveguides with moving media

The dyadic Green's function for cylindrical waveguides of circular or rectangular cross section with a moving, isotropic, homogeneous medium is developed using the method of eigenfunction expansion. The orthogonality properties of the vector mode functions are discussed. In contrast to waveguides with a stationary medium, it is seen that the normalization factor in the case of the E mode introduces a pole in the integral representation for the Green's function which must be excluded from the integration contour.     

Two-dimensional Green's functions in anisotropic multiferroic bimaterials with a viscous interface

We derive, by virtue of the unified Stroh formalism, the extremely concise and elegant solutions for two-dimensional and (quasi-static) time-dependent Green's functions in anisotropic magnetoelectroelastic multiferroic bimaterials with a viscous interface subjected to an extended line force and an extended line dislocation located in the upper half-plane. It is found for the first time that, in the multiferroic bimaterial Green's functions, there are 25 static image singularities and 50 moving image singularities in the form of the extended line force and extended line dislocation in the upper or lower half-plane. It is further observed that, as time evolves, the moving image singularities, which originate from the locations of the static image singularities, will move further away from the viscous interface with explicit time-dependent locations. Moreover, explicit expression of the time-dependent image force on the extended line dislocation due to its interaction with the viscous interface is derived, which is also valid for mathematically degenerate materials. Several special cases are discussed in detail for the image force expression to illustrate the influence of the viscous interface on the mobility of the extended line dislocation, and various interesting features are observed. These Green's functions can not only be directly applied to the study of dislocation mobility in the novel multiferroic bimaterials, they can also be utilized as kernel functions in a boundary integral formulation to investigate more complicated boundary value problems where multiferroic materials/composites are involved.     

Thermoelectroelastic Green's function and its application for bimaterial of piezoelectric materials

Summary For a two-dimensional piezoelectric plate, the thermoelectroelastic Green's functions for bimaterials subjected to a temperature discontinuity are presented by way of Stroh formalism. The study shows that the thermoelectroelastic Green's functions for bimaterials are composed of a particular solution and a corrective solution. All the solutions have their singularities, located at the point applied by the dislocation, as well as some image singularities, located at both the lower and the upper half-plane. Using the proposed thermoelectroelastic Green's functions, the problem of a crack of arbitrary orientation near a bimaterial interface between dissimilar thermopiezoelectric material is analysed, and a system of singular integral equations for the unknown temperature discontinuity, defined on the crack faces, is obtained. The stress and electric displacement (SED) intensity factors and strain energy density factor can be, then, evaluated by a numerical solution at the singular integral equations. As a consequence, the direction of crack growth can be estimated by way of strain energy density theory. Numerical results for the fracture angle are obtained to illustrate the application of the proposed formulation. Received 10 November 1997; accepted for publication 3 February 1998     

Fundamental solutions in antiplane elastodynamic problem for anisotropic medium under moving oscillating source

In this paper we study the antiplane problem of concentrated point force moving with constant velocity and oscillating with constant frequency in unbounded homogeneous anisotropic elastic medium.The explicit representation of the elastodynamic Green's function is obtained by using Fourier integral transform techniques for all rates of source motion as a sum of the integrals over the finite interval. The dynamic and quasistatic components of the Green's function are extracted. The stationary phase method is applied to derive an asymptotic approximation at the far wave field. The simple formulae for Poynting energy flux vectors for moving and fixed observers are presented too.It is shown that the motion brings some differences in the far field properties, such as, for example, fast and slow waves appearance under superseismic motion and modification of the wave propagation zones and their numbers.The case of isotropic medium is considered separately. For isotropic material all main formulae are obtained in explicit forms.     

Scattering of water waves by thin vertical plate submerged below ice-cover surface

The present paper is concerned with scattering of water waves from a vertical plate, modeled as an elastic plate, submerged in deep water covered with a thin uniform sheet of ice. The problem is formulated in terms of a hypersingular integral equation by a suitable application of Green's integral theorem in terms of difference of potential functions across the barrier. This integral equation is solved by a collocation method using a finite series involving Chebyshev polynomials. Reflection and transmission coefficients are obtained numerically and presented graphically for various values of the wave number and ice-cover parameter.     

GREEN＇S FUNCTIONS OF TWO－DIMENSIONAL ANISOTROPIC BODY WITH A PARABOLIC BOUNDARY

ＧＲＥＥＮ＇ＳＦＵＮＣＴＩＯＮＳＯＦＴＷＯ－ＤＩＭＥＮＳＩＯＮＡＬＡＮＩＳＯＴＲＯＰＩＣ ＢＯＤＹ ＷＩＴＨ Ａ ＰＡＲＡＢＯＬＩＣ ＢＯＵＮＤＡＲＹ（胡元太）（赵兴华）ＧＲＥＥＮ＇ＳＦＵＮＣＴＩＯＮＳＯＦＴＷＯ－ＤＩＭＥＮＳＩＯＮＡＬＡＮＩＳＯＴＲＯＰ?..     

HALF-SPACE GREENS'S FUNCTION DUE TO A SPATIALLY HARMONIC LINE LOAD AND SCATTERED FIELDS IN THE FAR FIELD REGION

Half-space Green's function due to a spatially harmonic line load has been expressed asa sum of the full-space Green's functions and a 2-D integral representation of the reflected waves bythe free surface of the half-space.By using the obtained half-space Green's function,an integral rep-resentation of the scattered waves by a cylindrical obstacle is then derived.Finally,by analyzing thefar-zone behavior of the integrands of the integral representation.the far-field pattern of the scatteredwaves in a half-space obtained.     

饱和土埋置力源的三维动力Lamb问题解答

基于一组弹性土波动方程,应用Fourier级数展开和Hankel积分变换,得到了三维问题饱和土骨架与孔隙水的应力及位移分量在变换域内的积分形式通解考虑地基表面透水情形,由边界条件导出了半空间饱和土体在埋置力源作用下的三维动力Lamb问题的解答给出了埋置水平力作用下地基表面竖向位移、径向位移及周向位移的数值解该研究为运用边界元法求解饱和地基的动力响应课题奠定了理论基础.     

Complete eigenfunction expansion form of the Green's function for elastic layered half-space

Summary  The complete eigenfunction expansion form of the Green's function for a 3-D elastic layered half-space in the frequency domain is derived in this paper. The expression of the Green's function presented here is an extension of that represented by the residue terms and the branch line integrals given by Lamb [1]. The present expression, however, clarifies the mathematical common frame between the residue terms and the branch line integrals with respect to the eigenfunctions and energy integrals. For the derivation, the concept of an energy integral for the improper eigenfunctions is newly developed. The improper eigenfunctions, which can be found in the wavenumbers for the branch cuts, are not in L ₂ space, so the definition of the energy integral requires some treatment. The energy integral is defined as the limit of the inner product of the improper eigenfunction and the definition function of the improper eigenfunction, for which the inner product remains finite. Via the definition of the energy integral, the kernel of the branch line integral is decomposed into the improper eigenfunction, and the complete eigenfunction expansion form of the Green's function is derived. The Green's function can thus be expressed by summation of normal modes, complex modes pointed out in [2], the integral of the improper eigenfunction and the residue at k=0 due to the singularity of the horizontal wavefunction. Received 3 May 2001; accepted for publication 23 August 2001     

Integration of Singularities in FE/BE Analyses of Soil-foundation Interaction with Non-homogeneous Elastic Soils

This paper addresses the derivation of the boundary integral equations for a non-homogeneous elastic half-space subjected to constant surface tractions on an arbitrarily shaped area on the basis of the respective Green's functions. The type of non-homogeneity considered is a power law variation of Young's modulus with depth below the surface of the half-space. Two different methods, a contour integral and an integration-free approach are presented, applicable for arbitrarily and rectangular shaped boundary elements, respectively. In the first one the divergence theorem is used in order to reduce the integration of a two-dimensional surface element to an integration over the element's confining boundary only. In the second approach no integration at all is needed since the solution is found simply by evaluating functions to be determined at the boundaries of the loaded rectangle.     

Application of the Local Boundary Integral Equation Method to Boundary-Value Problems

A review of the meshless formulations based on local boundary integral equation (LBIE) methods is presented. Physical quantities are approximated by the moving least-squares method. A summary of recent developments in the application of the LBIE method to potential problems, elastostatics, elastodynamics, thermoelasticity, and plate bending problems is given. The efficiency and generality of the present formulation in a wide class of engineering problems are confirmed.     

Reflection and transmission of obliquely incident Rayleigh surface waves by a viscoelastic interphase

The reflection and transmission of obliquely incident Rayleight surface waves by an interphase between two quarter spaces of identical or different materials, have been investigated. The mechanical behavior of the interphase is represented by a thin viscoelastic layer. By using the full space Green's functions due to a spatially harmonic line load, the mathematical statement of the 3-dimension problem is reduced to a 2-dimension system of singular integral equations. The far-field behavior of the scattered waves leads to the definition of reflection and transmission coefficients,R andT. The system of the singular integral equations are solved forR andT with the boundary element method. The results are presented for selected values of the elastic constants of the joined quarter spaces, the parameters of the interphase and the incident angles of Rayleigh surface waves.     

A Radiation Condition for Layered Elastic Media

This study aims to establish a generalized radiation condition for time-harmonic elastodynamic states in a piecewise-homogeneous, semi-infinite solid wherein the “bottom” homogeneous half-space is overlain by an arbitrary number of bonded parallel layers. To consistently deal with both body and interfacial (e.g. Rayleigh, Love and Stoneley) waves comprising the far-field patterns, the radiation condition is formulated in terms of an integral over a sufficiently large hemisphere involving elastodynamic Green's functions for the featured layered medium. On explicitly proving the reciprocity identity for the latter set of point-load solutions, it is first shown that the layered Green's functions themselves satisfy the generalized radiation condition. By virtue of this result it is further demonstrated that the entire class of layered elastodynamic solutions, admitting a representation in terms of the single-layer, double-layer, and volume potentials (distributed over finite domains), satisfy the generalized radiation condition as well. For a rigorous treatment of the problem, fundamental results such as the uniqueness theorem for radiating elastodynamic states, Graffi's reciprocity theorem for piecewise-homogeneous domains, and the integral representation theorem for semi-infinite layered media are also established.     

Some functional inequalities,with application to Green's function

Some elementary inequalities for functional with functional derivatives of given sign are presented and proved. These are then applied to the study of Green's functions for diffusion processes in a medium with sources and sinks (or alternately to Green's functions for the Schrödinger operator). The resulting inequalities are shown to include as quite special cases the super and sub-additive inequalities of potential theory. One consequence of the generalization is that scattering length is also shown to have sub-additive properties.This research was supported in part by the National Science Foundation and the U.S. Atomic Energy Commission.     

A boundary element method for calculating the SH waves scattering from arbitrarily shaped holes in an anisotropic medium

The scattering problem of elastic wave by arbitrarily shaped cavities in an infinite anisotropic medium is investigated by the boundary integral equation (BIE) method. The formulations of BIE are derived with the help of generalized Green's formula. The discretization of BIE is based upon constant elements. After confirmation of the accuracy of the present method, some numerical examples are given for various cavities in a full space, in which an isotropic body with a circular cylinder hole is used for comparison and good agreement is observed. It has been proved that the method developed in this paper is effective.     

平面问题等价边界积分方程的三次边界轮廓法

基于弹性力学平面问题等的边界积分方程,给出了三次单元的边界轮廓法。根据平面问题解的复变函数表示,构造了三次形函数。给出了对于混合边值问题求解系统方程确定的边界轮廓方程配置和三次单元界轮廓法的实施。     

Construction of asymptotic solutions to scattering problems in the fourier transform representation

Summary It is shown that the construction of a high-frequency asymptotic solution for the problem of a slit can be greatly simplified in the Fourier transform representation. This is accomplished by iterative solution of a pair of coupled integral equations whose kernels are the Green's functions for two isolated half planes.Part of the results given in this paper were presented at the USNC-URSI Spring Meeting held in Washington, D. C. in May, 1964.The author was with the Department of Electrical Engineering, New York University, when the research reported here was performed.     

Axisymmetric boundary-value problems in a piezoelectric medium

Based on the general solution of three-dimensional problems in piezoelectric medium, with the method of Green's functins^[2], axisymmetric boundary-value problems are discussed. The purpose of this research is for analyzing the effective on mechanics and electricity of the piezoelectric ceramics caused by voids and inclusions. The displacement, traction and electric Green's functions corresponding to circular ring loads acting in the interior of a piezoelectric ceramic are obtained. A cylindrical coordinate system is employed and Hankel transform are applied with respect to radial coordinates. Explicit solutions for Green's functions are presented in terms of infinite integrals of Lipshitz-Hankel type. By solving a traction boundary-value problem, the solution scheme is illustrated. Supported by the National Natural Science Foundation of China and the Foundation of the Open Laboratory of Solid Mechanics.     

A review of boundary and finite element methods in fracture mechanics

Reviewed in this work are the methods of finite and boundary element as applied to solve fracture mechanics problems. The former requires the discretization of the interior of the domain while the latter involves computing an integral equation over the boundary of the domain. Applications of these methods are made to two-dimensional elastic crack problems. Efficiency and accuracy of different approaches are discussed and compared by examples. The boundary element procedure employing special Green's functions for the plane crack problem is shown to be superior. The correlation between the hybrid element formulations and boundary element regions embedded into a finite element model is also given.     

同震断层三维裂纹扩展波动时域超奇异积分法

应用波动时域超奇异积分法将P波、S波和磁电热弹多场耦合作用下同震断层任意形状三维裂纹扩展问题转化为求解以广义位移间断率为未知函数的超奇异积分方程组问题;定义了广义应力强度因子,得到裂纹前沿广义奇异应力增量解析表达式;应用波动时域有限部积分概念及体积力法,为超奇异积分方程组建立了数值求解方法,编制了FORTRAN程序,以三维矩形裂纹扩展问题为例,通过典型算例,研究了广义应力强度因子随裂纹位置变化规律;分析了同震断层裂纹扩展中力、磁、电场辐射规律.