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.     

各向异性体内含任意孔洞对反平面波散射的边界元方法

本文借助于广义格林公式导出了用位移表示的各向异性介质中SH波入射时的边界积分方程.根据本文作者在文献[8]给出的基本解,求解了各向异性介质中孔洞对SH波的散射问题.边界积分方程的离散基于常数元模式.文中给出了一个圆柱、一个椭圆柱和两个椭圆柱形式的孔洞周围的位移场和应力场的数值结果.最后,对入射波频率较高时的情形作了说明.     

A non-primitive boundary element technique for modeling flow through non-deformable porous medium using Brinkman equation

In this study, we develop a non-primitive boundary integral equation (BIE) method for steady two-dimensional flows of an incompressible Newtonian fluids through porous medium. We assume that the porous medium is isotropic and homogeneous, and use Brinkman equation to model the fluid flow. First, we present BIE method for 2D Brinkman equation in terms of the non-primitive variables namely, stream-function and vorticity variables. Subsequently, a test problem namely, the lid-driven porous cavity over a unit square domain is presented to assert the accuracy of our BEM code. Finally, we discuss an application of our proposed method to flows through porous wavy channel, which is a problem of significant interest in the micro-fluidics, biological domains and groundwater flows. We observe that the rate of convergence (\(R_{c}\)) increases with increasing Darcy number. For low Darcy number streamlines follow the curvature of the wavy-walled channel and no circulation occurs irrespective of the wave–amplitude, while for high Darcy number the flow circulation occurs near the crest of the wavy-walled channel, when the wave–amplitude is large enough.     

Fracture Analysis for a Frictional Fastener Hole with Edge Cracks in an Anisotropic Plate∗

Abstract

Stress intensity factors are evaluated for a singly or doubly cracked fastener hole with frictional traction in an anisotropic plate, using a special kernel boundary integral equation (BIE) approach. The integration kernel (Green's function) used in this BIE approach has already taken the presence of the crack (or cracks) into account, thus.avoiding the need for element discretization to model the stress singularity at the crack tip. The Green's function employed is that of an infinite anisotropic plate containing an elliptical hole or crack, and subjected to an arbitrarily positioned point force. Several types of normal and shear traction conditions at the pinhole interface are considered. Numerical results are obtained for various geometrical and loading conditions and are compared with known solutions, where available, for their isotropic counterparts.     

Scattering of guided waves by circumferential cracks in composite cylinders

A somewhat generalized numerical procedure is used in this paper to study the problem of wave scattering by circumferential cracks in composite pipes. The study is motivated by the need to develop a model for the quantitative, ultrasonic non-destructive evaluation of cracks in pipes. For this purpose, a stiffness-based Rayleigh–Ritz type approach is employed first to obtain the approximate wave numbers and wave modes. Using the wave function expansions of the incident and scattered fields in the axial direction and decomposing the problem into separate symmetric and anti-symmetric problems, a three-dimensional wave scattering problem is reduced to two, independent two-dimensional problems over the circular cross-section. Both these problems can be reduced further to quasi-one-dimensions by discretizing the cross-section into finite elements and using a transfer matrix approach in the circumferential direction. This simplification greatly reduces the computational time. A comparison of the results for an isotropic pipe demonstrates the reliability and accuracy of the modified numerical procedure. Numerical results for the reflection and transmission coefficients of different incident wave modes are also presented for a 2-ply composite pipe with a crack. The crack may have an arbitrary circumferential length and radial depth. Simple extrapolations from one wave to another wave, separately incident on a crack, are demonstrated to be impossible due to different mode conversions by the crack.     

Lamb wave scattering by a symmetric pair of surface-breaking cracks in a plate

The scattering problem of a Lamb wave incident on a symmetric pair of surface-breaking transverse cracks in a plate is considered. The Lamb wave is assumed to be obliquely incident on the crack plane. Since the cracks are part-through, the scattered field will contain reflected as well as transmitted waves. The energy of the incoming wave is partitioned into reflected and transmitted wave modes. Energy coefficients of the reflected and transmitted waves are calculated as a function of incident frequency and crack depth. The incidence angle of the incoming wave is also treated as a parameter. Both the reflected and transmitted wave fields are considered as linear superpositions of all real and complex wave modes in the plate. Decomposition of modes is achieved with the help of an orthogonality condition based on the principle of reciprocal work. Continuity of displacement and stress fields is imposed at the crack plane. Energy coefficients for reflection and transmission are obtained from the mode amplitudes. Energy coefficients are shown to be a strong function of incident frequency and crack depth. Experiments are conducted with a PZT transducer network interacting with a symmetric pair of machined cracks in an aluminum plate. Trends predicted by the analysis are reflected in the experimental results.     

Cracks’ closure in 3-D fracture dynamics: the effect of relative location of two coplanar cracks

In this paper, the problem of two equal coplanar cracks with allowance for the crack faces contact interaction was investigated. The problem of the cracks located in homogeneous, isotropic, and linearly elastic solid subjected to normally incident tension–compression wave is solved by the boundary integral equations method. The influence of the distance between two cracks on the stress intensity factors (opening mode and transverse shear mode) is studied for a range of wave numbers. The results are compared with those obtained neglecting cracks’ closure.     

SHEAR WAVE SCATTERING FROM A PARTIALLY DEBONDED PIEZOELECTRIC CYLINDRICAL INCLUSION

I. INTRODUCTION Owing to the intrinsic coupling characteristics between electric and elastic behaviors, piezoelectricmaterials have been used widely in technology such as transducers, actuators, sensors, etc. Studieson electroelastic problems of a piezo…     

弹性理论几类导数边界积分方程之间的变换关系

导数场边界积分方程通常难以应用，因为存在着超奇异主值积分的计算障碍。弹性理论中有几类不同的位移导数边界积分方程，本文采用算子δij和∈ij(排列张量)作用于这些导数边界积分方程，做一系列变换，原有的超奇异积分被正则化为强奇异积分获解。从而建立了这些位移导数边界积分方程之间的转换关系，它们均可以归结为自然边界积分方程。自然边界积分方程仅存在容易计算的Cauchy主值积分。自然边界积分方程分析可直接获得边界应力和位移导数。     

Multiple reciprocity boundary element analysis of two-dimensional anisotropic thermoelasticity involving an internal arbitrary non-uniform volume heat source

In the direct boundary element method (BEM) formulation of anisotropic thermoelasticity, thermal loads manifest themselves as additional volume integral terms in the boundary integral equation (BIE). Conventionally, this requires internal cell discretisation throughout the whole domain. In this paper, the multiple reciprocity method in BEM analysis is employed to treat the general 2D thermoelasticity problem when the thermal loading is due to an internal non-uniform volume heat source. By successively performing the “volume-to-surface” integral transformation, the general formulation of the associated BIE for the problem is derived. The successful implementation of such a scheme is illustrated by three numerical examples.     

BIEM analysis of dynamically loaded anti-plane cracks in graded piezoelectric finite solids

Anti-plane cracks in finite functionally graded piezoelectric solids under time-harmonic loading are studied via a non-hypersingular traction based boundary integral equation method (BIEM). The formulation allows for a quadratic variation of the material properties in two directions. The boundary integral equation (BIE) system is treated by using the frequency dependent fundamental solution based on Radon transforms. Its numerical solution provides the displacements and tractions on the external boundary as well as the crack opening displacements from which the mechanical stress intensity factor (SIF) and the electrical displacement intensity factor (EDIF) are determined. Several examples for single and multiple straight and curved cracks demonstrate the applicability of the method and show the influence of the different system parameters.     

Diffraction of an antiplane shear wave by two coplanar griffith cracks in an infinite elastic medium

The scattering of a time-harmonic antiplane shear wave by two parallel and coplanar Griffith cracks embedded in an infinite elastic medium is considered. The input wave normally impinges on the cracks. Fourier transformations are utilized to reduce the problem to two simultaneous integral equations which can be solved by the series expansion method. The dynamic stress intensity factors are numerically computed.     

Solution of multiple crack problem in a finite plate using coupled integral equations

This paper studies a numerical solution of multiple crack problem in a finite plate using coupled integral equations. After using the principle of superposition, the multiple crack problem in a finite plate can be converted into two problems: (a) the multiple crack problem in an infinite plate and (b) a usual boundary value problem for the finite plate. For the former problem, the Fredholm integral equation is used. For the latter problem, a BIE based on complex variable is suggested in which a Cauchy singular kernel exists. For the proposed BIE, after using the inverse matrix technique, the dependence of the traction at a domain point from the boundary tractions is formulated indirectly. This is a particular advantage of the present study. Several numerical examples are provided and the computed results for stress intensity factor and T-stress at crack tips are given.     

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

This article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic problems in semi-infinite media. The main concepts underlying fast elastodynamic BEMs and the kernel-dependent elastodynamic FM-BEM based on the diagonal-form kernel decomposition are reviewed. An elastodynamic FM-BEM based on the half-space Green’s tensor suitable for semi-infinite media, and in particular on the fast evaluation of the corresponding governing double-layer integral operator involved in the BIE formulation of wave scattering by underground cavities, is then presented. Results on numerical tests for the multipole evaluation of the half-space traction Green’s tensor and the FMM treatment of a sample 3D problem involving wave scattering by an underground cavity demonstrate the accuracy of the proposed approach. The article concludes with a discussion of several topics open to further investigation, with relevant published work surveyed in the process.     

SH波对有部分脱胶衬砌的圆形孔洞的散射

本文研究了圆形孔洞内衬砌与孔洞部分脱胶时对ＳＨ波的散射．将脱胶区看作表面不相接触的弧形界面裂纹，利用波函数展开法，并引入裂纹面的位错密度函数，将问题归结为一组奇异积分方程．通过数值计算获得了动应力强度因子（ＤＳＩＦ）和远场位移及散射截面（ＳＣＳ）．结果显示：由于脱胶，ＤＳＩＦ和ＳＣＳ在较低的频率上发生共振．     

Two Cracks Emerging from a Single Point,under the Influence of a Longitudinal Shear Wave

The problem of determining the dynamic stress intensity coefficients for two cracks emerging from a single point is solved. The cracks are affected by a longitudinal shear wave. The original problem is reduced to solving a system of two singular integro-differential equations with fixed singularities. For an approximate solution of this system, a numerical method is proposed that takes into account the real asymptotics of the unknown functions and uses special quadrature formulas for singular integrals.     

各向异性体对瞬态SH波散射问题的边界元方法

本文利用Fourier变换和加权残数法建立了正交各向异性体反平面瞬态波散射问题的边界积分方程,构造了在边界上能满足波动方程的边界元函数,它比常用的二次元和高次元具有较好的适应性。数值结果表明利用该形函数计算高频情形下的散射问题具有很高的精度。     

有限散射信号下二维缺陷形状识别的罚函数方法

研究在有限照射角度和频带宽度下二维缺陷的形状识别问题。首先,通过引进介质参数扰动函数,建立介质参数扰动函数和弹性波散射场之间的非线性关系,并将所关心的缺陷的形状识别问题转化为关于扰动函数的反演;然后,利用变分技术和优化方法求解,为了弥补散射数据的不足,在总的目标函数中,采用附加度量函数作为罚函数;最后,对后场散射远场测量时有限照射角度和频带宽度下几种典型缺陷进行了模拟识别,表明了;表明了罚函数法的有效性。     

Degenerate scale for multiply connected Laplace problems

The degenerate scale in the boundary integral equation (BIE) or boundary element method (BEM) solution of multiply connected problem is studied in this paper. For the mathematical analysis, we use the null-field integral equation, degenerate kernels and Fourier series to examine the solvability of BIE for multiply connected problem in the discrete system. Two treatments, the method of adding a rigid body term and CHEEF concept (Combined Helmholtz Exterior integral Equation Formulation), are applied to remedy the non-unique solution due to the critical scale. The efficiency and accuracy of the two regularizations are also addressed. For simplicity without loss of generality, the eccentric case is considered to demonstrate the occurring mechanism of degenerate scale.