A Filtered Backpropagation Algorithm for Diffraction Tomography

The concept of a relaxation spectrum is used to compute the absorption and dispersion of a linear anelastic solid. The Boltzmann after-effect equation is solved for a solid having a linear relationship between stress and strain and their first time derivatives, the ‘standard linear solid’, and having a distribution of relaxation times. The distribution function is chosen to give a nearly constant Q over the seismic frequency range. Both discrete and continuous relaxation spectra are considered. The resulting linear solid has a broad absorption band which can be interpreted in terms of a superposition of absorption peaks of individual relaxation mechanisms. 

The accompanying phase and group velocity dispersion imply that one cannot directly compare body wave, surface wave, and free oscillation data or laboratory and seismic data without correcting for absorption. The necessary formalism for making these corrections is given. In the constant Q regions the correction is the same as that implied in the theories of Futterman, Lomnitz, Strick and Kolsky.

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Velocity dispersion due to anelasticity; implications for seismology and mantle composition

The determination of the elastic properties of composite materials (multiphase aggregates, polycrystals, and porous or cracked solids) from the elastic properties of the components may be approached in several ways. The problem may be treated statistically, via scattering theory, through variational principles, or by the assumption of specific geometries for the material under consideration. Each of these methods is reviewed in turn. The widely used Voigt-Reuss-Hill average can be a poor approximation for both two-phase composites and polycrystals, and its replacement by the two Hashin-Shtrikman bounds is recommended. For pore-containing or crack-containing media, specific geometry models must be considered if useful results are to be obtained. If aggregate theory is used to estimate the moduli of individual components of a composite whose bulk properties are known, the shear moduli of the component phases must be matched (within a factor of 2 or 3) for the method to be useful. Results for nonlinear composites (which allow calculation of the pressure variation of aggregate moduli) have been obtained for only a few special cases.

The Elastic Properties of Composite Materials

Keratin: Structure, mechanical properties, occurrence in biological organisms, and efforts at bioinspiration

General electromechanical reciprocity relations applied to the calculation of elastic wave scattering coefficients

An integral equation is used to derive formal expressions for the scattering of a plane wave from a single homogeneous flaw embedded in an isotropic elastic medium. Expressions are found for the scattered amplitudes and differential cross sections. An optical theorem is also derived.

Formal aspects of the theory of the scattering of ultrasound by flaws in elastic materials

We used the integral equation formulation of the scattering of elastic waves to generate an approximate solution analogous to the Born approximation in quantum mechanics. This solution is attractive because of the ease with which it may be applied to scatterers of complicated shapes. We investigated the validity of the approximation by comparing it with exact results for spherical scatterers. Our conclusion for voids in elastic media is that the approximation describes well the scattering when the wavelength of the incident wave is approximately an order of magnitude larger than the scatterer and when the scattering is viewed in the backscattered directions. For many applications this range of validity is experimentally accessible. For elastic inclusions, however, where the properties of defect and host differed by 20–40%, the Born approximation is surprisingly good for all angles and even at short wavelengths.

The Born approximation in the theory of the scattering of elastic waves by flaws

We calculate the effective elastic properties of various polycrystals and rocks using a self-consistent integral equation technique developed in the theory of disordered alloys. This technique includes but is more general than the self-consistent approximation developed by Hershey, Kroner (1958), and others. In practice the technique is easily adaptable to computer analysis for modest cost. Our computed results compare well with experimental data. Also we point out that an approximation for the perfectly disordered aggregate developed by Kroner (1967) is not exact as claimed but is an approximation to the selfconsistent approximation.

The elasticity of polycrystals and rocks

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J. E. Gubernatis

Papers

Formal aspects of the theory of the scattering of ultrasound by flaws in elastic materials

The Born approximation in the theory of the scattering of elastic waves by flaws

The elasticity of polycrystals and rocks

Fundamental theory of elastic wave scattering by defects in elastic materials: integral equation methods for application to ultrasonic flaw detection

Theory of the Scattering of Ultrasound by Flaws