Inversion of elastic waveform data in anisotropic solids using the delta-function representation of the Green’s function
27 Oct 1998-Journal of the Acoustical Society of America (Acoustical Society of America)-Vol. 104, Iss: 3, pp 1716-1719
TL;DR: In this article, a method for inversion of measured data on elastic waveforms in anisotropic solids is proposed, using the delta-function representation of the Green's function.
Abstract: A method for inversion of measured data on elastic waveforms in anisotropic solids is proposed, using the delta-function representation of the Green’s function. The method requires integration over a closed 2-D (two-dimensional) space. In contrast, inversion using the traditional Fourier representation requires integration over an infinite 4-D space. The method can be used to determine the Green’s function for imaging applications and elastic constants for materials characterization. The method is illustrated by applying it to determine all six elastic constants of a model graphite fiber composite assuming a tetragonal structure and using simulated data.
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TL;DR: The proposed computational algorithms are programmed in a MATLAB environment by incorporating symbolic calculations performed using Maple Computer Algebra System, and demonstrate the accuracy of the proposed algorithm in evaluating the Green's functions.
Abstract: Time-harmonic Green's functions for a triclinic anisotropic full-space are evaluated through the use of a symbolic computation system.This procedure allows evaluation of the Green's functions for the most general anisotropic materials. The proposed computational algorithms are programmed in a MATLAB environment by incorporating symbolic calculations performed using Maple Computer Algebra System. Extensive testing of the numerical results has been performed for both displacement and stress fields. The tests demonstrate the accuracy of the proposed algorithm in evaluating the Green's functions. Copyright © 2001 John Wiley & Sons, Ltd.
26 citations
TL;DR: In this article, a method for numerical evaluation of three-dimensional time-harmonic Green's functions for nonisotropic media is proposed, which avoids repeated evaluation of the common parts of the integrands by dealing with entire elastodynamic state vector.
17 citations
TL;DR: In this paper, an indirect boundary integral equation approach was used to estimate the surface motion of a basin of arbitrary shape embedded in a half-space using point sources distributed on the so-called auxiliary surfaces, with their intensities determined from the requirement that the boundary and the continuity conditions are to be satisfied in the least-squares sense.
Abstract: Scattering 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 satisfied 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 finite integrals over a unit sphere or a unit circle which can be numerically evaluated very efficiently. 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 defined 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 significant amplification of surface motion atop the basin. While the amplitude of peak surface motion may be similar to the corresponding isotropic results, the difference in the displacement patterns may be quite different between the two. Therefore, this study clearly demonstrates that material anisotropy may be very important for accurate assessment of surface ground motion amplification atop basins.
14 citations
TL;DR: In this paper, the authors examined the characteristics of elastic wave propagation in viscoelastic porous media, which contain simultaneously both the Biot-flow and the squirt-flow mechanisms (BISQ).
6 citations
Additional excerpts
...…to derive representation integrals for the radiation from an arbitrary distribution of body forces and surface tractions, and can be applied to acoustic theory and numerical computations of many fields (Tewary, 1998; Spies, 1997; Liu et al., 2002; Liu and Zhang, 2001; Dravinski and Zheng, 2000)....
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References
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TL;DR: In this paper, acoustic resonance spectroscopy was used to determine the complete elastic constants of a uniaxial fiber-reinforced metal-matrix composite: boron-aluminum.
Abstract: Acoustic‐resonance spectroscopy was used to determine the complete elastic constants of a uniaxial‐fiber‐reinforced metal‐matrix composite: boron‐aluminum. This material exhibits orthotropic macroscopic symmetry and, therefore, nine independent elastic stiffnesses Cij. The off‐diagonal elastic constants (C12,C13,C23), which contain large errors when measured by conventional methods, were especially focused on. Good agreement emerged among present results, a previous pulse‐echo study, and theoretical predictions using a plane‐scattered‐wave ensemble‐average model. Attempts to measure the internal‐friction ‘‘tensor’’ failed.
61 citations
TL;DR: In this article, the Radon transform is used to reduce the set of 3D hyperbolic partial differential equations to a set of 1-dimensional pde's, and the 3D solution is subsequently expressed as an inversion integral, in the form of an integral over a unit sphere.
52 citations
TL;DR: A computationally efficient representation of the three-dimensional elastostatic and elastodynamic Green's functions for anisotropic solids is derived by solving the Christoffel equation in terms of delta functions, which is also applicable to other wave equations.
Abstract: A computationally efficient representation of the three-dimensional elastostatic and elastodynamic Green's functions for anisotropic solids is derived by solving the Christoffel equation in terms of delta functions. The representation is also applicable to other wave equations. The method is applied to calculate the transient and the static displacement field due to a point source in infinite and semi-infinite anisotropic cubic solids. For elastodynamic calculations in anisotropic solids, our representation saves the computational time by a factor of about 1000 over the conventional Fourier-Laplace representation. In the elastostatic case, the computational efficiency of our method is much more than the conventional Fourier representation but comparable to the methods of Barnett and Barnett and Lothe in specific cases. I. INTRQDUCTIGN We derive an integral representation for the elastodynarnic Green's function in terms of a 5 function. In this representation, even in the general anisotropic 3D (three-dimensional} case, only a 1D integration needs to be done numerically. The integration does not involve oscillatory functions or singularities. This reduces the problem of numerical convergence. The CPU time required for calculating the elastodynamic Green's function is only about, ' of that required for the conventional Fourier-Laplace representation. ' Qur technique is also applicable to other wave equations such as the electromagnetic or acoustic wave equation. We also calculate the elastostatic Green's function by taking the static limit of the elastodynamic response of a solid. Qur method is much more efticient than the conventional Fourier representation. However, the computational eKciency of our method is comparable to that of Barnett" for an infinite solid and to that of Barnett and Lothe for calculations of surface displacements in a semi-infinite solid. The elastodynamic Green's function is useful for calculating physical properties of solids involving longwavelength phonons. Presently, there is a strong interest in wave-form-based ultrasonics for nondestructive characterization of anisotropic materials. These techniques measure the response of a material to an elastic pulse. which is very well modeled in terms of the elastodynamic Green's function. The elastostatic Green's function is useful for calculating stress distribution in solids containing defects and discontinuities. ' In 1attice statics calculations and the computer simulation of lattice defects, the elastostatic Green's function is needed to fix the asymptotic limit of the lattice distortions. ' The Green's function for an isotropic solid is usually' calculated by using the Fourier-I. aplace representation in the wave-vector — frequency space. This is adequate for traditional materials which could be approximated as isotropic. Modern composite materials are highly anisotropic. In this case the conventional Fourier representation requires a 4D integration over oscillatory functions for the elastodynamic Green's function and 3D integration for the elastostatic Green's function. In many applications, we need to calculate the Green's function for bounded solids (containing free surfaces of interfaces). The Fourier-Laplace representation of the Green's function in such cases is poorly convergent and its evaluation is CPU intensive. A powerful technique for calculating the 3D elastostatic Green's function was suggested by Barnett. This technique is very well suited for calculation of the Green's function for infinite solids. " For this case, the computational e%ciency of our method is about the same as that of Barnett. An extension of Stroh's formalism has been developed by Barnett and Lothe for bounded solids, but it is applicable only to 2D problems. Qur representation retains its simplicity even for 3D bounded solids. For example, our expression for the Green's function for a semi-infinite solid satisfies the free-surface boundary con
40 citations
TL;DR: In this paper, the delta function representation of the elastodynamic Green's function is applied to calculate the three-dimensional waveforms for elastic-pulse propagation at the free surface in a half-space tetragonal solid.
Abstract: The delta‐function representation of the elastodynamic Green’s function is applied to calculate the three‐dimensional waveforms for elastic‐pulse propagation at the free surface in a half‐space tetragonal solid.
9 citations