Theory of scattering from multilayered bodies of arbitrary shape

Green 's functions and boundary integral equation methods are used to derive a matrix set of equations for scattering from a multilayered homogeneous elastic body embedded in an infinite elastic material. The surfaces separating the layers have arbitrary shape. The formalism for the three-layer material is derived in detail and generalized to N-layers. A matrix factorization method (MFM) is shown to considerably simplify the computational problem. The relation to the problems of acoustic waves in fluids and electromagnetic waves in a dielectric material is briefly indicated.     

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.     

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.     

Scattering of elastic waves by non-planar cracks

A modification of the null field approach is used to study the scattering of elastic waves by non-planar cracks. A fictitious surface is added to the crack so that a convenient closed surface is obtained and the surface fields on this closed surface are expanded in vector spherical harmonics. The edge conditions are introduced into these expansions and this is shown to be essential for the numerical convergence. Total cross sections and backscattering amplitudes as functions of frequency are computed numerically for rotationally symmetric cracks which are part of spherical or spheroidal surfaces. By integration in frequency backscattered pulses are also computed. Some cases with two cracks are also considered.     

Wave scattering by a floating porous elastic plate of arbitrary shape: A semi-analytical study

In this paper, a semi-analytical model based on linear potential flow theory and an eigenfunction expansion method is developed to study wave scattering by a porous elastic plate with arbitrary shape floating in water of finite depth. The water domain is divided into the interior and exterior regions, corresponding to the domain beneath the plate and the rest extending towards infinite distance horizontally, respectively. The unknown coefficients in the potential expressions are determined by satisfying the continuity conditions for pressure and velocity at the interface of the two regions, together with the conditions for the motion/force at the edge of the plate, where the Fourier series expansion method is employed to deal with the terms associated with the radius function. A plate with three shapes – circular, cosine and elliptical – and three edge conditions are considered. We find that the largest deflection of the plate with a simply supported edge and a clamped edge is more sensitive to the change in porosity when the porosity is small. The power dissipated by an elliptical plate with its major axis perpendicular to the incident wave direction is the largest among the case studies for the majority of the porosity values tested.     

Asymptotic elastic solution for two straight cracks of arbitrary length and location

An infinite elastic plane containing two straight cracks of arbitrary length and location is analyzed within the framework of elastostatics. The mathematical formulation is based on the stress solution for a single crack and leads to a system of singular integral equations that govern the crack surface displacement densities. The solution series in terms of the reciprocal of the crack centre distance is not suitable for cracks that are spaced too closely. It is shown by way of examples that the method of asymptotic solution is convenient for developing approximation expressions of the stress and displacement field with certain characteristics. The formulas for the stress intensity factors and crack opening are given for the case of a constant tensile load. Graphical results are given for the variations of the stress intensity factors with parameters depending on the relative positions of the cracks.     

Surface and interfacial waves of arbitrary form in isotropic elastic media

A method due to Friedlander of accommodating disturbances of arbitrary form into the theory of surface waves in a semi-infinite isotropic elastic body is extended and shown to yield a simple closed form solution for the displacement field. An analogous treatment of interfacial waves of arbitrary form at a plane contact discontinuity separating different isotropic elastic materials is also given.

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Résumé On développe une méthode, conçue par Friedlander, qui fait entrer les perturbations de forme arbitraire dans la théorie des ondes de surface dans un corps élastique isotropique semi-infini, et on montre qu'elle permet d'obtenir une solution simple et exacte pour le champ de déplacement. Les ondes de forme arbitraire qui existent dans le plan à la frontière de materiaux élastiques isotropiques differents sont traitées de façon analogue.

     

Interaction of harmonic elastic waves with closely situated arbitrarily oriented cracks

The interaction of plane harmonic elastic waves with closely situated arbitrarily oriented cracks is considered. The contact pressure forces of the edges and friction in their contact area are taken into account. An algorithm for numerical iteration solution is constructed and some effects of the mutual influence of cracks are investigated. Special Design and Engineering Office, Physicotechnical Institute of Low Temperatures, Kharkov, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 61–67, October, 1999.     

AN EXACT METHOD OF BENDING OF ELASTIC THIN PLATES WITH ARBITRARY SHAPE

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Theory of water waves in an elastic vessel

Recent experiments related to the Dragon Wash phenomena showed that axisymmetric capillary waves appear first from excitation, and circumferential capillary waves appear after increase of the excitation strength. Based on this new finding, a theory of parametric resonance is developed in detail to explain the on-set of the prominent circumferential capillary waves. Numerical computation is also carried out and the results agree generally with the experiments. Analysis and numerical computation are also presented to explain the generation of axisymmetric low-frequency gravity waves by the high-frequency external excitation.     

Interaction of elastic waves with a periodic array of collinear inplane cracks

A boundary integral equation method is applied to the study of the interaction of plane elastic waves with a periodic array of collinear inplane cracks. Numerical results are presented for the dynamic stress intensity factors. The effects of the wave type, wave frequency, wave incidence angle, and crack spacing on the dynamic stress intensity factors are analyzed in detail. The project supported by the Committee of Science and Technology of Shanghai and Tongji University     

A numerical model for the nonlinear interaction of elastic waves with cracks

Microstructures such as cracks and microfractures play a significant role in the nonlinear interactions of elastic waves, but the precise mechanism of why and how this works is less clear. Here we simulate wave propagation to understand these mechanisms, complementing existing theoretical and experimental works.We implement two models, one of homogeneous nonlinear elasticity and one of perturbations to cracks, and then use these models to improve our understanding of the relative importance of cracks and intrinsic nonlinearity. We find, by modeling the perturbations in the speed of a low-amplitude P-wave caused by the propagation of a large-amplitude S-wave that the nonlinear interactions of P- and S-waves with cracks are significant when the particle motion is aligned with the normal to the crack face, resulting in a larger magnitude crack dilation. This improves our understanding of the relationship between microstructure orientations and nonlinear wave interactions to allow for better characterization of fractures for analyzing processes including earthquake response, reservoir properties, and non-destructive testing.     

The scattering of harmonic elastic anti-plane shear waves by two collinear cracks in the piezoelectric plate by using the non-local theory

In this paper, the dynamic interaction between two collinear cracks in a piezoelectric material plate under anti-plane shear waves is investigated by using the non-local theory for impermeable crack surface conditions. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. These equations are solved using the Schmidt method. This method is more reasonable and more appropriate. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The project supported by the Natural Science Foundation of Heilongjiang Province and the National Natural Science Foundation of China(10172030, 50232030)     

The scattering of harmonic elastic anti-plane shear waves by two collinear cracks in anisotropic material plane by using the non-local theory

In this paper, the dynamic behavior of two collinear cracks in the anisotropic elasticity material plane subjected to the harmonic anti-plane shear waves is investigated by use of the nonlocal theory. To overcome the mathematical difficulties, a one-dimensional nonlocal kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress field near the crack tips. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near crack tips. The nonlocal elasticity solutions yield a finite hoop stress at the crack tips, thus allowing us to using the maximum stress as a fracture criterion. The magnitude of the finite stress field not only depends on the crack length but also on the frequency of the incident waves and the lattice parameter of the materials.