Multiple scattering theory for heterogeneous elastic continua with strong property fluctuation: theoretical fundamentals and applications

Abstract

Scattering of elastic waves in heterogeneous media has become one of the most important problems in the field of wave propagation due to its broad applications in seismology, natural resource exploration, ultrasonic nondestructive evaluation and biomedical ultrasound.Nevertheless, it is one of the most challenging problems because of the complicated medium inhomogeneity and the complexity of the elastodynamic equations. A widely accepted model for the propagation and scattering of elastic waves, which properly incorporates the multiple scattering phenomenon and the statistical information of the inhomogeneities is still missing. In this work, the author developed a multiple scattering model for heterogeneous elastic continua with strong property fluctuation and obtained the exact soluti on to the dispersion equation under the first-order smoothing approximation. The model establishes an accurate quantitative relation between the microstructural properties and the coherent wave propagation parameters and can be used for characterization or inversion of microstructures. Starting from the elastodynamic differential equations, a system of integral equation for the Green functions of the heterogeneous medium was developed by using Green's functions of a homogeneous reference medium. After properly eliminating the singularity of the Green tensor and introducing a new set of renormalized field variables, the original integral equation is reformulated into a system of renormalized integral equations. Dyson's equation and its first-order smoothing approximation, describing the ensemble averaged response of the heterogeneous system, are then derived with the aid of Feynman's diagram technique. The dispersion equations for the longitudinal and transverse coherent waves are then obtained by applying Fourier transform to the Dyson equation. The exact solution to the dispersion equations are obtained numerically. To validate the new model, the results for weak-property-fluctuation materials are compared to the predictions given by an improved weak-fluctuation multiple scattering theory. It is shown that the new model is capable of giving a more robust and accurate prediction of the dispersion behavior of weak-property-fluctuation materials.Numerical results further show that the new model is still able to provide accurate results for strong-property-fluctuation materials while the weak-fluctuation model is completely failed. As applications of the new model, dispersion and attenuation curves for coherent waves in the Earth's lithosphere, the porous and two-phase alloys, and human cortical bone are calculated. Detailed analysis shows the model can capture the major dispersion and attenuation characteristics, such as the longitudinal and transverse wave Q-factors and their ratios, existence of two propagation modes, anomalous negative dispersion, nonlinear attenuation-frequency relation, and even the disappearance of coherent waves. Additionally, it helps gain new insights into a series of longstanding problems, such as the domi...

Multiple Scattering Theory for Polycrystalline Materials with Strong Grain Anisotropy: Theoretical Fundamentals and Applications

“…Following the standard procedure as comprehensively presented in [51], we obtain the integral representation of the perturbed field, i.e., the displacement and strain Green's functions of the polycrystalline material composed of an infinite number of randomly oriented crystallites, as follows:…”

“…As rigorously analyzed in [51], the integral in the second term on the righthand side of Eq. (7) has δ singularities due to the second order derivative of the homogeneous Green's function.…”

“…(7) has δ singularities due to the second order derivative of the homogeneous Green's function. By introducing the concept of shape-dependent principal value [51], the correct definition of the matrix D is given by:…”

“…Similar to the elastic stiffness fluctuation δcij, the renormalized elastic variable Ξpq is also dependent on the orientation of each grain, i.e., it is a function of the Euler angles. The governing equation for the resemble average of the stochastic variable, i.e., the coherent response of the scattered field, can be obtained using the Feynman diagram technique, for which the general procedure is presented in the author's previous work [51]. If further invoking the first-order-smoothing approximation [44,51,[53][54]…”

“…In this work, the author strives to develop a universal multiple scattering theory for polycrystalline materials with general anisotropic crystallites, regardless of whether the grains are weakly anisotropic. Following the new paradigm created in the previous work [51], the renormalized Dyson's equation for polycrystalline materials are derived in Section II, by using the Fourier transform technique, the closed form solution of the coherent wave field in spacetime is obtained. Meanwhile the dispersion equation for the longitudinal and transverse coherent waves are derived.…”