Three-dimensional time-harmonic elastodynamic Green
            <sup>’</sup>
            s functions for anisotropic solids

Wang, C.-Y.; Achenbach, J. D.

doi:10.1098/rspa.1995.0052

Cited by 119 publications

(67 citation statements)

References 19 publications

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“…The explicit expressions of the Green's functions, can be found in the paper by Wang and Achenbach [11]. If the unknown density functions are assumed in the form of discrete point sources, i.e.…”

Section: Solution Of Problemmentioning

confidence: 99%

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Scattering of elastic waves by a 3D anisotropic basin

Zheng¹,

Dravinski²

2000

Earthquake Engng. Struct. Dyn.

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SUMMARYScattering of elastic waves by a three-dimensional transversely isotropic basin of arbitrary shape embedded in a half-space is considered using an indirect boundary integral equation approach. The unknown scattered waves are expressed in terms of point sources distributed on the so-called auxiliary surfaces. The sources are expressed in terms of the full-space Green's functions with their intensities determined from the requirement that the boundary and the continuity conditions are to be satis"ed in the least-squares sense. Steady-state results were obtained for incident plane pseudo-P-, SH-, SV-, and Rayleigh waves.Using the Radon transform the Green's functions are obtained in the form of "nite integrals over a unit sphere or a unit circle which can be numerically evaluated very e$ciently.Detailed analysis of the method includes the discussion on the shape of the auxiliary surfaces and the distribution of the collocation points and sources. The convergence criteria is de"ned in terms of transparency tests, isotropic limit test, and minimization of a certain norm. The isotropic limit tests show excellent agreement with the isotropic results available in literature.For anisotropic materials the numerical results are given for a semispherical basin. The results show that presence of an anisotropic basin may result in signi"cant ampli"cation of surface motion atop the basin. While the amplitude of peak surface motion may be similar to the corresponding isotropic results, the di!erence in the displacement patterns may be quite di!erent between the two. Therefore, this study clearly demonstrates that material anisotropy may be very important for accurate assessment of surface ground motion ampli"cation atop basins.

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Section: Solution Of Problemmentioning

confidence: 99%

“…Recently, several researchers approached the solution of the problem of the Green's functions in an anisotropic medium by using the Radon transform [7,11]. The Radon transform has been studied extensively in computational tomography [12].…”

Section: Introductionmentioning

confidence: 99%

Scattering of elastic waves by a 3D anisotropic basin

Zheng¹,

Dravinski²

2000

Earthquake Engng. Struct. Dyn.

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“…Therefore, this specific representation provides the zero-order waveguide modes (27). Let us compare this expression to the Radon transform of u(r) (28), which is defined as u R ͑s, n͒ ϭ ͵ dru͑r͒␦͑͑s Ϫ n ⅐ r͒͒. [ B 4 ] In Eq.…”

Section: Appendix A: Helmholtz Decomposition Developmentmentioning

confidence: 99%

Determination and analysis of guided wave propagation using magnetic resonance elastography

Romano

Abraham

Rossman

et al. 2005

Magnetic Resonance in Med

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We present a novel extension of standard magnetic resonance elastography (MRE) measurement and analysis methods, which is applicable in cases where the medium is characterized by waveguides or fiber bundles (i.e., muscle) leading to constrained propagation of elastic waves. As a demonstration of this new method, MRI is utilized to identify the pathways of the individual fibers of a stalk of celery, and 3D MRE is then performed throughout the volume containing the celery fibers for a measurement of the displacements. A Helmholtz decomposition is performed permitting a separation of the displacements into longitudinal and transverse components, and an application of a hybrid Radon transform permits a spectral decomposition of wave propagation along the fibers. Dot product projections between these elastic displacements measured in the global coordinate system and three Frenet vectors representing the tangent and two corresponding orthogonal vectors along any particular fiber orientation yield the displacement contributions to wave propagation along the fiber as if it were a waveguide. A sliding window spatial Fourier transform is then performed along the length of each fiber to obtain dispersion images that portray space-wavenumber profiles.

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“…In the last decade some articles reported the synthesis of full-space or half-space solutions using the Radon transform. A fundamental solution for 3D anisotropic full-space using the Radon transform was presented by Wang and Achenbach [4]. In this article [4], the fundamental solutions for the 3D problem are given in terms of numerical integrations to be performed over a unit sphere.…”

Section: Introductionmentioning

confidence: 99%

“…A fundamental solution for 3D anisotropic full-space using the Radon transform was presented by Wang and Achenbach [4]. In this article [4], the fundamental solutions for the 3D problem are given in terms of numerical integrations to be performed over a unit sphere. A numerical realization scheme for the full-space anisotropic solution was presented by Dravinski and Niu [5].…”

Section: Introductionmentioning

confidence: 99%

Numerically evaluated displacement and stress solutions for a 3D viscoelastic half space subjected to a vertical distributed surface stress loading using the Radon and Fourier transforms

Adolph

Mesquita

Carvalho³

et al. 2006

Commun. Numer. Meth. Engng.

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SUMMARYIn this article a numerical solution for a 3D isotropic, viscoelastic half-space subjected to vertical rectangular surface stress loading of constant amplitude is evaluated with the aid of the Radon and Fourier integral transforms. Dynamic displacement and stress fields are computed for the half-space surface as well as for points inside the domain. The analysis is performed in the frequency domain. Viscoelastic effects are incorporated by means of the elastic-viscoelastic correspondence principle. The equations of motion are solved in the Radon-Fourier transformed domain. Inverse transformations to the physical domain are accomplished numerically. The scheme used to perform the numerical inverse transformations is addressed. The solution is validated by comparison with results available in the literature. A sample of original dynamic results for displacement and stress fields for the 3D half-space is furnished.

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Three-dimensional time-harmonic elastodynamic Green ^’ s functions for anisotropic solids

Cited by 119 publications

References 19 publications

Scattering of elastic waves by a 3D anisotropic basin

Scattering of elastic waves by a 3D anisotropic basin

Determination and analysis of guided wave propagation using magnetic resonance elastography

Numerically evaluated displacement and stress solutions for a 3D viscoelastic half space subjected to a vertical distributed surface stress loading using the Radon and Fourier transforms

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Three-dimensional time-harmonic elastodynamic Green ’ s functions for anisotropic solids

Cited by 119 publications

References 19 publications

Scattering of elastic waves by a 3D anisotropic basin

Scattering of elastic waves by a 3D anisotropic basin

Determination and analysis of guided wave propagation using magnetic resonance elastography

Numerically evaluated displacement and stress solutions for a 3D viscoelastic half space subjected to a vertical distributed surface stress loading using the Radon and Fourier transforms

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Product

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Three-dimensional time-harmonic elastodynamic Green ^’ s functions for anisotropic solids