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

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

Huijing He† Department of Mechanical and Materials Engineering University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA † he.hui.jing@hotmail.com

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 solution 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 dominant mechanism of seismic attenuation and the existence of the Mohorovičić discontinuity. This work provides a general and accurate theoretical framework for quantitative characterization of microstructures in a broad spectrum of heterogeneous materials and it is anticipated to have vital applications in seismology, ultrasonic nondestructive evaluation and biomedical ultrasound. I. INTRODUCTION Scattering of elastic waves is the central topic in the theory of elastodynamics and its applications cover a broad spectrum of physical and engineering disciplines, ranging from seismology, natural resource exploration, ultrasonic nondestructive evaluation to biomedical ultrasound. Multiple scattering is a universal phenomenon which occurs in any materials that exhibit spatial fluctuation of elastic moduli or density. The planet Earth is a typical heterogeneous media. Geophysical studies show the rocks constituting the lithosphere, typically with a characteristic size from several hundred meters to tens of kilometers, exhibit strong fluctuations in both the elastic stiffness and the density. A seismic wave gets scattered by large amounts of such inhomogeneities during its travel from the source, normally tens of kilometers beneath the Earth’s surface, to the observation station located up to thousands of kilometers away from the epicenter. As a consequence, the seismic signal experiences multiple scattering and appears as a wavetrain consists of several dispersive and attenuating 2

direct arrivals followed by a long duration of coda waves. Multiple scattering

…(Full text truncated)…