Interaction of plane SH-waves with the tips of interface cracks

On the basis of the solution of the problem of diffraction of SH-waves on the interface crack obtained earlier, we study the specific features of the far radiated field for very small and critical sounding angles. The effects of multiple diffraction of waves on the crack tips and the formation of the scattered field of displacements are analyzed. It is shown that, in these cases, the dependence of the amplitude of the field on the crack length for very small and critical angles of irradiation is monotonic. This fact can be used for the determination of the crack sizes. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 2, pp. 92–98, March–April, 2008.     

SH-waves at a corrugated interface between a dry sandy half-space and an anisotropic elastic half-space

Summary A problem of reflection and transmission of a plane SH-wave incident at a corrugated interface between a dry sandy half-space and an anisotropic elastic half-space is investigated. Rayleigh's method of approximation is applied to derive the reflection and transmission coefficients for the first and second order approximation of the corrugation. The expressions for reflection and transmission coefficients for the first order approximation of the corrugation are obtained in closed form for a special type of interface, and the energy partition relation is derived. It is found that these coefficients are proportional to the amplitude of corrugation and are functions of elastic properties of the half-spaces and also of the angle of incidence. Numerical examples illustrating the effects of the sandiness, the anisotropy, the corrugation of the interface, the frequency, and the angle of the incidence on the coefficients are presented.     

Scattering of inhomogeneous wave by viscoelastic interface crack

Summary The reflection, refraction and scattering of inhomogeneous plane waves of SH type by an interface crack between two dissimilar viscoelastic bodies are investigated. The singular integral equation method is used to reduce the scattering problem into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Then, the singular integral equation is solved numerically by Kurtz's piecewise continous function method. The crack opening displacement and dynamic stress intensity factor characterizing the scattered near-field are estimated for various incident angles, frequencies and relaxation times. The differences on crack opening displacement and stress intensity factor between elastic and viscoelastic interface crack are contrasted. And the effects of incident angle, incident frequency and relaxation time of the viscoelastic material are analyzed and explained by the features of phase lag and energy dissipation of the viscoelastic wave.     

Scattering of antiplane shear wave by a piezoelectric circular cylinder with an imperfect interface

Summary We present analytical solutions for the scattering of an antiplane shear wave by a piezoelectric circular cylinder with an imperfect interface. We first consider the simple case in which the imperfection is homogeneous along the interface. Two typical imperfect interfaces are addressed: 1) mechanically compliant and dielectrically weakly conducting interface, and 2) mechanically compliant and dielectrically highly conducting interface. The expressions for the directivity pattern and scattering cross-section of the scattered shear waves are derived. We then investigate the more difficult problem in which the imperfection is circumferentially inhomogeneous along the interface. A concise expression for an inhomogeneously compliant and weakly conducting interface is derived by means of matrix notation. Numerical examples are presented to demonstrate the effect of the imperfection and the circumferential inhomogeneity of the interface on the directivity patterns and scattering cross-sections of the scattered shear wave. The circumferentially inhomogeneous interface is also utilized to model the interface where an arbitrary number of cracks exist. Results show that when every part of the interface is rather compliant, large low-frequency peaks of the scattered cross-sections, which correspond to the resonance scattering, can be observed no matter if the interface is homogeneous or inhomogeneous. The appearance of large low-frequency peaks can be well explained by estimating the natural frequency of the corresponding reduced mass-spring system where the cylinder is assumed as a rigid body. Peaks of the scattered cross-sections spanning from low frequencies to high frequencies can be observed for a cylinder with a partially debonded interface.     

Diffraction of SH-waves on interface cracks under the action of concentrated forces

In the mathematically rigorous statement, we solve the problem of diffraction of elastic SH-waves on a finite (semiinfinite) crack located on the plane boundary of two perfectly joined materials. As a sounding field, we consider the field of a point source. The problem is reduced to the Wiener – Hopf functional equation, which is solved by the method of factorization. An approximate solution of this equation is obtained. We numerically study the distribution of the field in the radiation zone and the distinctive features of its formation depending on the location of the source of radiation. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 6, pp. 67–77, November–December, 2008.     

Stress field in a crack on the interface of materials irradiated with SH-waves

On the basis of the solution of the problem of diffraction of a field of plane SH-waves on a crack located on the interface of materials obtained by the authors earlier, we deduce asymptotic expressions for the stress intensity factors at the crack tips. The dependences of the stress intensity factors on the dimensionless crack length, the angle of incidence of the plane wave, and the mechanical and physical properties of the materials are investigated. The possibility of appearance of singularities of the stress intensity factors near the remote crack tip (relative to the source of radiation) for the tangential and critical angles of irradiation is demonstrated. It is shown that, for the critical angles of incidence, these singularities can appear only in perfect materials and that, in the presence of losses, the increase in the stress intensity factors is bounded. It is also demonstrated that the indicated effects are caused by the diffraction interaction of SH-waves with the crack tips. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 43, No. 4, pp. 18–30, July–August, 2007.     

Interaction of SH-waves by a Griffith crack in an infinite transversely isotropic medium

As a follow-on to a previous problem involving imposed P-waves, the diffraction of SH-waves by a finite Griffith crack located on a symmetric surface of an infinite transversely-isotropic medium is considered. A Fredholm integral equation is derived for the determination of diffracted field. From the integral equation length symptotic development of the solution is obtained which is valid for wavelength compared to the crack length. For wavelengths comparable with the size of the crack, the integral equation is solved numerically. The stress-intensity factor and displacement field in the vicinity of the crack are compared for a range of values of frequency. The approximate solution is compared with the exact solution.

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Résumé Comme suite à l'étude d'un problème précédent, dans lequel on imposait un régime d'ondes de type P, on a considéré la diffraction d'ondes SH par une fissure finie de Griffith située sur la surface symétrique d'un milieu transversalement isotrope. On a déduit une équation intégrale de Fredholm pour déterminer le champ de diffraction. On en a tiré une solution en développement asymptotique qui est valable lorsque la longueur d'onde est grande par rapport à la longueur de la fissure. Pour des longueurs d'onde comparables à la longueur de la fissure, on résoud l'équation intégrale par voie numérique. On compare le facteur d'intensité d'entaille et le champ de déplacement dans le voisinage de la fissure pour diverses valeurs de fréquence.La solution approchée est comparée à la solution exacte.

     

Water-wave Scattering by an Ice-strip

The problem of water-wave scattering by a strip of ice-cover floating on the surface of deep water is investigated here. The ice-cover is modelled as a thin elastic plate of very small thickness. The problem is reduced to that of solving two singular integral equations of Carleman type over a semi-infinite range and are solved approximately by casting them into two separate Riemann-Hilbert problems by assuming the strip width to be large. The reflection and transmission coefficients are derived approximately. Numerical results for the reflection coefficient are presented graphically against the wave number and also against the ice-cover parameter. The oscillatory nature of the reflection coefficient against the wave number as well as the ice-cover parameter is found to be one of the main features of the curves. It is also seen that, in the limiting case when the ice-cover parameter tends to zero (i.e., the ice-cover is almost absent), the amount of reflection is negligible     

Transient SH-waves in an elastic plate with a moving edge

The propagation of transient SH-waves in a plate is discussed. An SH-wave source is placed on a fixed edge of the plate and the other edge moves outward with uniform velocity. Two Laplace transforms in fixed and moving coordinate systems are employed. Developing an exchange formula between two Laplace transforms, a coupled functional equation for two unknown coefficients is derived and solved by iteration. The solution obtained here is exact up to three reflection times of an incident (source) wave, and the time-development of the wave front shape is also discussed.     

Scattering of torsional waves by a spherical cavity in a long circular elastic cylinder

Summary. The axisymmetric wave equation is solved for the problem of torsional elastic waves scattered by a spherical cavity located symmetrically in an infinitely long circular cylinder. Using Fourier transforms, the problem is reduced to the solution of an infinite system of simultaneous equations, which is suitable for the numerical solution. The numerical results on the transmission and reflection coefficients are shown for various values of sphere radius and frequency. Equation of energy conservation is utilized to check the numerical procedure.     

Scattering of SH wave by a crack terminating at the interface of a bimaterial

A general method for solving the scattering of plane SH wave by a crack terminating at the interface of a bimaterial is presented. The crack can terminate at the interface in an arbitrary angle. In order to solve the proposed problem, the Greens function for a point harmonic force applied at an arbitrary point of the bimaterial is established by the Fourier transformation method. Using the obtained Greens function and the Betti-Rayleigh reciprocal theorem, the total scattered field of the crack is constructed. The total scattered field of the crack is divided into a regular part and a singular part. The hypersingular integral equation of the crack is obtained in terms of the regular and singular scattered field as well as the free wave field. The stress singularity order and singular stress at the terminating point are analyzed by the hypersingular integral equation and the singular scattered field of the crack. The dynamic stress intensity factor (DSIF) at the terminating point is defined in terms of the singular stresses at the terminating point. Numerical solution of the hypersingular integral equation gives the DSIFs at the crack tips. Comparison of our results with known results confirms the proposed method. Some numerical results and corresponding analysis are given in the paper.Constructive advice from the anonymous reviewers is acknowledged.     

Scattering of elastic waves by an elastic sphere

The problem of scattering of plane compressional wave by an elastic sphere embedded in an isotropic elastic medium of different material properties is solved. Approximate formulas are derived for the displacement field, stress tensor, stess intensity factors, far-field amplitudes and the scattering cross section. It is assumed that the wave length is large compared to the radius of the scatterer. Various elastostatic limits are also presented.     

Scattering of elastic waves by an elastic sphere

The problem of scattering of plane compressional wave by an elastic sphere embedded in an isotropic elastic medium of different material properties is solved. Approximate formulas are derived for the displacement field, stress tensor, stress intensity factors, far-field amplitudes and the scattering cross-section. It is assumed that the wave length is large compared to the radius of the scatterer. Various elastostatic limits are also presented.     

Scattering of SH waves by an arc-shaped interface crack between a cylindrical magneto-electro-elastic inclusion and matrix with the symmetry of 6 mm

Summary Investigated are the scattered field properties of SH waves by a magneto-electro-elastic cylindrical inclusion partially debonded from its surrounding magneto-electro-elastic material in this paper. The debonding region is modeled as an arc-shaped interface crack with magneto-electrically permeable surfaces. By expressing the scattered fields as the wave function expansions with unknown coefficients, the mixed boundary value problem is first reduced to a set of dual series equations. Then the dislocation density function is introduced as an unknown to transform the dual series equations to a Cauchy singular integral equation of the first type, which can be solved numerically. Finally, the scattered field intensity factor, the scattered far field pattern and scattered cross section are obtained. The effects of incident direction, crack opening angle and material combination on the properties of both near fields and far fields are respectively examined graphically. The solution of this problem is relevant to ultrasonic nondestructive detection of the debonding between two dissimilar magneto-electro-elastic materials and is expected to have applications to the question of how dynamic loading can lead to growth of debonds.     

Scattering of Scalar Waves by an Impenetrable Rough Sphere

Scattering of scalar waves by an impenetrable, slightly rough sphere is considered. The incoherent differential cross-section is calculated to the lowest order in hk (where k is the wavenumber and h ² the mean-square deviation from a smooth sphere), but for any value of the size parameter α = kR₀ (where R ₀ is the radius of the sphere). Particular attention is paid to the back-scattering cross-section σ_B for large α, under the assumption that the correlation length of the roughness structure is not larger than the incident wavelength. The resulting limit of σ_B is well known: the same cross-section is obtained by approximating the sphere locally by a tangential plane and using results for slightly rough planar areas.     

On the relationship between the stress intensity factors and the far field of SH-waves diffracted on interface cracks

By using the Wiener–Hopf method, we solve the problem of diffraction of plane SH-waves on semiinfinite interface cracks and deduce the relationships between the radiation field in fixed directions and the stress intensity factors. The dependence of the stress intensity factors on the field of displacements scattered in the plane perpendicular to the crack tip is analyzed. It is shown that the ratio of the stress intensity factors can be represented via the ratio of the scattered far fields measured for various angles and frequencies of sounding.     

Scattering of a Gaussian beam by an impedance half-plane

The diffraction of a Gaussian beam by an impedance half-plane is studied through the method of the modified theory of physical optics. An electric line source, which is defined in the complex space, is used to represent the Gaussian beam. The uniform evaluation of the diffraction integral is performed and the scattering patterns of the field are investigated for various numerical parameters of the incident wave.     

Trifurcation of a crack due to plane SH-waves in an infinite elastic medium

The dynamic anti-plane problem of trifurcation of a semi-infinite crack due to incidence of two linearly varying plane SH-waves with non-parallel wave fronts in an infinite elastic medium has been considered. The semi-infinite crack is assumed to trifurcate when the plane waves intersect the crack tip. The problem has been solved using the self-similar technique, which is based on the observation that certain field variables show dynamic similarities. The results include the expressions for shear stress in the planes of the cracks and the stress intensity factors at the crack tips. Numerical calculations have been carried out to show the variations of stress intensity factors at the crack tips with the angle of skew for different values of the crack tip velocity and angle of incidence.     

Scattering of SH waves by an arbitrarily orientated closed crack

The 2D problem of a time-harmonic plane shear horizontal (SH) wave scattered by a finite closed crack in an isotropic material is presented in the paper. The crack is arbitrarily orientated with regard to the incident wave. A spring model based on the assumption that the traction components on the crack surfaces are linearly related to the crack opening displacement (COD) is used to model the closed crack. The problem is formulated in a set of boundary integral equations which contains the CODs as unknowns. Numerical examples are presented for the CODs, elastodynamic stress intensity factors, and the scattered displacement field for various parameters, such as spring stiffness, crack sizes and crack orientations. The results show that both the crack closure and orientation have significant effects on the scattered displacement field for the closed crack.