Scattering and diffraction of SH waves by a finite crack: an analytical solution (original) (raw)
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The scattering of SH waves by a finite crack with a superposition-based diffraction technique
Studia Geophysica et Geodaetica, 2016
The problem of diffraction of cylindrical and plane SH waves by a finite crack is revisited. We construct an approximate solution by the addition of independent diffracted terms. We start with the derivation of the fundamental case of a semi-infinite crack obtained as a degenerate case of generalized wedge. This building block is then used to compute the diffraction of the main incident waves. The interaction between the opposite edges of the crack is then considered one term at a time until a desired tolerance is reached. We propose a recipe to determine the number of required interactions as a function of frequency. The solution derived with the superposition technique can be applied at low and high frequencies.
Diffraction of SH cylindrical waves by a finite crack: an analytical solution
Geophysical Journal International, 2010
A new analytic solution for the problem of a line source of harmonic SH waves that struck a finite 2-D plane crack is introduced. This construct is based on Macdonald's solution for the problem of cylindrical acoustic waves diffracted by a wedge. We consider this approach, since special cases of wedges include semi-infinite cracks. The finite crack is assembled by superimposing the effects of two semi-infinite cracks. The new solution is obtained by taking into account diffracted waves generated at the edges travelling along the faces of the crack. Repeated wave fields that result from fitting the two cracks are removed. Comparisons with the results obtained with the Indirect Boundary Element Method (IBEM) are shown. Both frequency and time domain cases are included.
Scattering of SH wave by a crack terminating at the interface of a bimaterial
Computational Mechanics, 2004
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 Green's function for a point harmonic force applied at an arbitrary point of the bimaterial is established by the Fourier transformation method. Using the obtained Green's 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.
Scattering and diffraction by a single crack: an accuracy analysis of the rotated staggered grid
Geophysical Journal International, 2005
Scattering and diffraction of elastic waves by many cracks or inclusions have a significant influence on the waves behaviour (e.g. velocity, attenuation). As a result of the lack of an analytical solution for scattering caused by complex crack structures, numerical methods have to be used to compute the wave field. The numerical error of such methods has to be controlled. In this paper, we present an accuracy analysis of the rotated staggered grid (RSG)-finite difference (FD) scheme. It is designed, amongst other things, to model wave propagation in multiple cracked rocks with high contrasts in their elastic moduli without implementing explicit boundary conditions and without averaging elastic moduli (e.g. at free surfaces like crack boundaries). The problem of a plane SH wave scattered by a single finite crack has a known analytical solution. This solution provides a chance to validate the special abilities of the RSG-FD scheme. The calculation precision is examined by comparing numerical and analytical results. We observe a very good agreement for the single finite crack. Because the RSG is not restricted to the single crack case only, we conclude that this approach is an accurate tool to study wave propagation in media with many cracks.
A semi-analytical approach in the high-frequency diffraction by cracks
Mechanics Research Communications, 2011
In the problem of high-frequency diffraction by cracks in linear elastic materials we propose a numerical method which is based on a separation of the oscillating (GTD) solution and a certain slowly varying function. This technique, described in literature for regular (Fredholm) integral equations, is applied here to hyper-singular equations arising in diffraction by thin cracks in elastic media. The algorithm proposed is efficient for both high and moderate frequencies.
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Geofísica Internacional, 2016
Scattering and diffraction of P and SV waves caused by parallel oriented cracks located near to a free surface are investigated in this work. The Indirect Boundary Element Method (IBEM) was applied for studying the wave propagation phenomena in a half-plane model that contain the cracks. Various incidence angles of P and SV waves are considered. Sometime before it has been reported that a near free-surface crack generates scattered surface waves whose amplitude spectra show conspicuous resonance peaks. Such effect has been attributed to local resonances originated in a virtual layer between the shallowest crack and the free surface. For our case of two parallel crack system, where cracks are located at different depths, the amplitude spectra show additional peaks, which can be associated with the presence of the second crack. Given similar sizes between these two cracks, the characteristic resonance frequency observed at the free surface corresponds mainly to the equivalent layer fo...
Diffraction coefficients of a semi-infinite planar crack embedded in a transversely isotropic space
Wave Motion, 2009
We have considered a semi-infinite crack embedded in a transversely isotropic medium and studied two special cases, one, in which the axis of symmetry is normal to the crack face and the wave incidence is arbitrary and another, in which the axis lies in the crack plane normal to the edge and the incident wave vector is also normal to the edge. The problem is of interest in Non-Destructive Evaluation, because austenitic steels that are found in claddings and other welds in the nuclear reactors are often modeled as transversely isotropic. In both of cases, we have expressed the scattered field in a closed form and computed the corresponding diffraction coefficients.
Journal of Computational Mathematics, 2010
The wave scattering problem by a crack Γ in R 2 with impedance type boundary is considered. This problem models the diffraction of waves by thin two-sided cylindrical screens. A numerical method for solving the problem is developed. The solution of the problem is represented in the form of the combined angular potential and single-layer potential. The linear integral equations satisfied by the density functions are derived for general parameterized arcs. The weakly singular integrals and the Cauchy singular integral arising in these equations are computed using a highly accurate scheme with a truncation error analysis. The advantage of the scheme proposed in this paper is, in one hand, the fact that we do not need the analytic property of the crack and we allow different complex valued surface impedances in both sides of the crack. In the other hand, we avoid the hyper-singular integrals. Numerical implementations showing the validity of the scheme are presented.