The time domain elastodynamic Kirchhoff approximation for cracks: The inverse problem

Obtaining the size, shape, and orientation of a crack from ultrasonic elastic wave scattering information is one example of the solution of an inverse problem. Here, we obtain the formal solution to this inverse problem for ideal, flat cracks using the Kirchhoff approximation for the scattered elastodynamic wavefield. Time and frequency domain verisions of the solution will be given, both in the general case and, in reduced form, for circular cracks. The time domain inverse formulation, in particular, will be shown to be equivalent to the method of projections, leading to a classical two-dimensional Radon transform. A method is also demonstrated for performing a constrained inversion to obtain the parameters of a flat crack that is assumed a priori to be of elliptical shape.     

Scattering of monochromatic elastic waves on a planar crack of arbitrary shape

Scattering of monochromatic elastic waves on an isolated planar crack of arbitrary shape is considered. The 2D-integral equation for the crack opening vector is discretized by Gaussian approximating functions. For such functions, the elements of the matrix of the discretized problem have forms of standard one-dimensional integrals that can be tabulated. For regular grids of approximating nodes, the matrix of the discretized problem has the Toeplitz structure, and the corresponding matrix–vector products can be calculated by the fast Fourier transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. Examples of calculations of crack opening vectors, dynamic stress-intensity factors, and differential cross-sections of circular (penny-shaped) and non-circular cracks for various incident wave fields are presented. For a penny-shaped crack and longitudinal incident waves normal to the crack plane, an efficient semi-analytical method of the solution of the scattering problem is developed. The results of both methods are compared in a wide frequency region of the incident field.     

Surface displacements due to elastic wave scattering by buried planar and non-planar cracks

The scattering of time-harmonic plane longitudinal, shear, and Rayleigh waves by a crack in two dimensions embedded in a semi-infinite homogeneous isotropic elastic half-space has been studied in this paper. Two problems have been considered: a straight crack and a Y-shaped crack. A hybrid numerical technique combining a multipolar representation of the scattered field in the half-space with the finite element method has been used to obtain the far-field displacements as well as the stress-intensity factors for the crack tips. Results for vertical displacement on the free surface of the half-space are presented in this paper.     

Analytical extension of Finite Element solution for computing the nonlinear far field of ultrasonic waves scattered by a closed crack

Nonlinear scattering of ultrasonic waves by closed cracks subject to contact acoustic nonlinearity (CAN) is determined using a 2D Finite Element (FE) coupled with an analytical approach. The FE model, which includes unilateral contact with Coulomb friction to account for contact between crack faces, provides the near-field solution for the interaction between in-plane elastic waves and a crack of different orientations. The numerical solution is then analytically extended in the far-field based on a frequency domain near-to-far field transformation technique, yielding directivity patterns for all linear and nonlinear components of the scattered waves. The proposed method is demonstrated by application to two nonlinear acoustic problems in the case of tone-burst excitations: first, the scattering of higher harmonics resulting from the interaction with a closed crack of various orientations, and second, the scattering of the longitudinal wave resulting from the nonlinear interaction between two shear waves and a closed crack. The analysis of the directivity patterns enables us to identify the characteristics of the nonlinear scattering from a closed crack, which provides essential understanding in order to optimize and apply nonlinear acoustic NDT methods.     

On the Spectrum of the Interior Transmission Problem in Isotropic Elasticity

The present work concerns with the investigation of the interior transmission problem, which is naturally associated to the inverse elastic scattering problem of determining the support of an isotropic homogeneous penetrable body from a knowledge of the time harmonic incident plane waves and the far-field patterns of the corresponding scattered wave-fields. Our approach combines a boundary integral formulation of the problem and a compact perturbation argument to establish the discreteness of the set of transmission eigenvalues and the well-posedness of the interior transmission problem under the most general assumptions on the elastic parameters of the underlying media.      

Numerical method for the shape reconstruction of a hard target

IntroductionAninverseproblemofconsiderableimportanceinvariousfieldsofengineeringtechnology ,suchasnondestructivetesting ,medicalimaging ,remotesensingandseismicimaging ,istodeterminetheshapeofascatteringobjectfromitsfar_fieldeffectsontheacousticscatteringwaves.However,thiskindofproblemisparticularlydifficulttosolvesinceitisbothnonlinearandill_posed[1].Fortunately ,therehavebeenseveralmethodsdevelopedforsolvingnumericallytheinverseproblemduringthelastdecade .Ofparticularimportancearenonlinearop…     

大延伸非均匀介质中地震波全弹性散射理论Ⅰ--弹性波单次散射理论

所描述的工作聚焦于大延伸非均匀介质中非均匀弹性地震波散射问题的研究.应用Born近似及等效源原理,推出了来自连续横向无界非均匀层的弹性散射波的通解.这一工作是解决大延伸非均匀介质的弹性地震波多次散射问题的基础.在上述通解的基础上,建立了适用于大延伸非均匀介质的全弹性散射理论.该理论可包容小尺度非均匀体、大延伸非均匀介质全弹性波单次弱散射理论及标量波单次弱散射理论,因此可视其为一个更为广义和统一的弱散射理论.     

High-frequency scattering of elastic waves from cylindrical cavities

The scattering of time-harmonic plane longitudinal elastic waves by smooth convex cylindrical cavities is investigated. The exact solution for a circle is evaluated for wavelengths of the same order as the radius, and the geometrical and physical elastodynamics approximations are shown to be inadequate. The application of Watson's transformation exhibits the various diffraction effects and the relative importance of each is assessed. Excellent approximations for the scattered far-field are obtained with a hybrid method, in which an approximation for the surface field is constructed from the creeping wave contributions and this is then used in an integral representation. A generalization, based on the Geometrical Theory of Diffraction, of the hybrid method to cavities of smooth convex cross-section is presented and applied to the specific case of an ellipse. The predictions of the hybrid method compare well with numerical results obtained by an eigenfunction expansion method.     

Compressional waves in fluid-saturated porous solid containing a penny-shaped crack

Diffraction of normal compression waves by a penny-shaped crack in a fluid-saturated porous medium is investigated. Two wave types are considered, namely, compressional wave of the first kind, and the second kind. The former, also known as fast wave, propagates primarily through the solid, whereas the latter or slow wave, propagates mainly in the fluid. Each wave propagates in the medium along with induced wave of the same type in the companion constituent of the material. Application of Biot’s theory in conjunction with integral transform technique reduces the problem to a mixed boundary-value problem whose solution is in turn governed by a Fredholm integral equation of the second kind. Near-field and far-field solutions are obtained in terms of the dynamic stress-intensity factor and the scattering cross section, respectively. They are of particular importance to the linear elastic fracture mechanics (LEFM) and in the scattering theory of elastic waves. The mode I stress-intensity factors are computed numerically for a set of selected material property values, and shown graphically for various mass density and viscosity-to-permeability ratios. The obtained results reveal significant impact of the presence of pore fluid upon the stress-intensity factors, both magnitudes and frequencies at their peak values. The influence of the fluid is also observed from the calculated scattering cross sections of the scattered far-field. Accuracy of the present solution procedure is verified by comparing the numerical results with existing results in the limiting case of dry elastic materials.     

Wave momentum and scattering of elastic waves by two-dimensional thin objects

The elastic wave momentum equation is applied to scattering of dilatational and shear waves by two-dimensional thin objects. It is shown that the sources of wave momentum are located at the edges of these objects. For a stress-zero crack or for a rigid inclusion there are two sources at each edge, for a fluid-filled crack there is just one. The scattered wave is expressed in terms of these sources. This reduces the number of independent variables by a factor two. An application to inverse scattering problems is also given.     

Scattering of elastic waves by a small inclined surface-breaking crack

I we examine the scattering of Rayleigh waves by an inclined two-dimensional plane surface-breaking crack in an isotropic elastic half-plane. We follow the method already introduced by the authors (A and W , 1992a, J. Mech. Phys. Solids 40, 1683) to obtain an analytical solution when the parameter , which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The procedure employed is the method of matched asymptotic expansions, which involves determining the scattered wave field both away from and near the crack. The outer solution is specialized from the general expansion in the first part of this work (A and W , 1992a, J. Mech. Phys. Solids 40, 1683), and the inner problem is exactly solved by the Wiener-Hopf technique. The displacement field and scattered Rayleigh waves are found uniformly to third order in , and concluding remarks are made about the general method as well as the results presented here.     

多个椭圆柱波浪力的一种解析解

波浪在大尺寸结构表面产生不可忽略的散射波, 该散射波在多柱体体系中继续传播, 并在同体系中的其他柱体上产生高次散射波. 本文基于椭圆坐标系和绕射波理论首先推导了波浪作用下椭圆单柱体产生的散射波压力公式, 随后考虑该散射波在多柱体系中的传播, 将其视为第二次入射波, 推导出柱体上第二次散射波压力公式, 同理可以推导出高次散射波压力公式, 最后得到椭圆多柱体波浪力解析解, 并用数值解验证了本文解析方法的正确性. 本文以双柱体和四柱体体系为例, 分析了不同参数(波数、净距、波浪入射角度等)下, 高次散射波对柱体上波浪作用的影响. 结果表明: 波数较大的情况下, 高次散射波引起柱体上的波浪力不能忽略; 结构间距较大的情况下, 虽然高次波的作用有减小的趋势但仍然明显; 高次散射波来自多个柱体对入射波的散射, 柱体数目的增加后, 高次波的影响会增加, 结构所受的高次波作用因参数变化而起的波动会变剧烈; 高次波对上游柱体波浪力的贡献较对下游柱体的贡献大.      

一种求解瑞利波散射问题的修正边界元方法

边界元法的一大优势是用于求解半空间等无限域问题,然而对于弹性波的传播问题,传统边界元法在采用全平面或全空间格林函数时,在截断边界处仍会产生虚假的反射回波,直接影响到散射场的求解准确性。因此,本文在传统边界元法基础上提出一种修正边界元法,用于计算无限大半平面中的弹性波场问题。该方法以瑞利波形式的远端散射场代替原本因截断而舍去的部分,通过互易定理建立单位瑞利波和全平面格林函数的积分方程,求得修正系数,并代入修正边界元矩阵,计算出瑞利波的散射场。为验证本文所提方法,文中将多个算例的结果与解析解对比,并用该方法计算了不同缺陷的散射场。这些对比结果表明,本文所提修正边界元法可准确求解瑞利波散射场,为基于表面波的缺陷反演问题研究提供了有效的正演途径。     

Scattering of elastic waves by an inclusion with an interface crack

A method of perturbation is used to derive an integral representation of the displacement field for the scattering of a plane wave from an inclusion with an interface crack. In the long-wave approximation it is shown that the solution of only an associated static problem is required and formal expressions are derived for the scattered far field amplitudes and scattering cross section. In the case of a cylindrical inclusion the solution of the associated static problem is then used to find in a closed form the corresponding expressions for plane incident P- and S-waves.     

Inverse scattering problem from an impedance crack via a composite method

In this paper we consider an inverse scattering problem from an open arc with impedance boundary conditions on both sides of the crack. Our aim is to recover both the impedance function and the unknown crack simultaneously from the far-field pattern with only one incident wave. Making the most out of the direct problem, a straightforward method of iterative nature is developed for the inverse problem. The ill-posedness of this problem is considered by incorporating the Tikhonov regularization. Numerical examples are provided at the end of the paper to show the feasibility of our method.     

Generalized Born scattering in anisotropic media

The Born scattering approximation has been widely used in seismology to study scattered waves, and to linearize the propagation problem for inversion. The standard Born theory requires the model be separated into a smooth, reference model and a perturbation. Scattering occurs from the pertubation. In the distorted Born approximation, when the reference model is inhomogeneous, the reference Green's functions are normally not known exactly, but the error in these Green's functions is rarely quantified. In this paper, we generalize Born scattering theory to include the errors in the Green's functions explicitly, and obtain scattering integrals from these errors. For forward modelling, there is no need to separate the model into a reference and perturbation part - approximate Green's functions in the true model can be used to calculate the scattered signals.

The theory is developed for inhomogeneous, anisotropic media. Asymptotic ray theory results are suitable approximate Green's functions for the generalized Born scattering theory. The error terms are simple, easily calculated and included in the scattering integrals. Various applications of generalized Born scattering theory have already appeared in the literature, e.g. quasi-shear ray coupling, and this paper is restricted to an improved and more complete theoretical development. Further applications will appear elsewhere.     

Analysis of crack extension in anisotropic materials based on local normal stress

The normal stress ratio theory is applied to predict crack extension behavior in center-notched unidirectional graphite-epoxy of arbitrary fiber axis orientation, subjected to arbitrary far-field planar loading. The theory is applied within analytical solutions for two infinite plate geometries: a plate with a sharp center crack, and a plate with an elliptical center flaw. A critical analytical case is identified suggesting that application of the theory within a stress solution modelling crack tip shape may increase the accuracy of crack growth direction predictions. Crack extension direction, location of crack extension, and critical stress predictions of the theory are compared to those obtained from experiments on specimens subjected to tensile, shear, and mixed-mode far-field loading. The comparison shows that, applied within each analytical solution, the normal stress ratio theory provides verifiable predictions of crack growth behavior. By modelling actual notch tip shape, the elliptical notch solution is able to provide accurate qualitative predictions of the origin of crack extension along the periphery of a cut notch tip in a way that the sharp crack analysis cannot. The sharp notch solution appears to provide slightly more accurate crack growth direction predictions, however. Also, in predicting critical applied far-field stresses, the sharp crack solution appears to exhibit a stronger ability to model subtle experimental trends.     

Arbitrary point force in arbitrarily oriented transversely isotropic body

The published traditional point force problem solutions usually orient axes of coordinates in such a way that plane xOy is parallel to the planes of isotropy. We consider here a general case: an arbitrary point force is applied inside a transversely isotropic space, with arbitrary axes orientation. We obtain the field of displacements and stresses in terms of contour integrals, which are computable, because the solution for the traditional case is known. Identification of the set of contour integrals, which look impossible to compute, is a necessary first step toward the solution of non-traditional contact and crack problems for arbitrarily oriented cracks and punches.     

Theory of scattering of elastic waves from flat cracks of arbitrary shape

A method, based on a boundary-integral representation of the elastic displacement, for calculating crack-opening-displacements on a flat crack of arbitrary shape and for incident elastic waves of arbitrary direction, polarization, and wavelength is developed and illustrated by application to Rayleigh scattering from two families of crack shapes. The crack-opening-displacement is expanded in a truncated complete set of functions on the crack surface. This transforms the boundary-integral representation into a matrix equation with rank three times the order of the truncation. This matrix equation has the properties that it can be expressed as the result of an extremum principle with respect to variations of the expansion coefficients of the crack-opening-displacement (thus converges as the truncation order increases) and the matrix kernel (which must be inverted) is positive definite. A conclusion drawn is that only very accurate experiments can distinguish a flat crack of general shape from a penny-shaped crack by long-wavelength elastic-wave scattering.     

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.