ARTICLE | doi:10.20944/preprints202007.0323.v1
Subject: Earth Sciences, Geophysics Keywords: scattering; heterogeneity; anisotropy; elastic waves; dispersion; attenuation; Mohorovičić discontinuity
Online: 15 July 2020 (09:01:02 CEST)
The geometry of mesoscopic inhomogeneities plays an important role in determining the macroscopic propagation behaviors of elastic waves in a heterogeneous medium. Non-equiaxed inhomogeneities can lead to anisotropic wave velocity and attenuation. Developing an accurate scattering theory to describe the quantitative relation between the microstructure features and wave propagation parameters is of fundamental importance for seismology and ultrasonic nondestructive characterization. This work presents a multiple scattering theory for strongly scattering elastic media with general tri-axial heterogeneities. A closed analytical expression of the shape-dependent singularity of the anisotropic Green’s tensor for the homogeneous reference medium is derived by introducing a proper non-orthogonal ellipsoidal coordinate. Renormalized Dyson’s equation for the coherent wave field is then derived with the help of Feynman’s diagram technique and the first-order-smoothing approximation. The exact dispersion curves and the inverse Q-factors of coherent waves in several representative medium models for the heterogeneous lithosphere are calculated numerically. Numerical results for small-scale heterogeneities with the aspect ratio varying from 1 to 7 show satisfactory agreement with those obtained from real earthquakes. The results for velocity dispersion give rise to a novel explanation to the formation mechanism of different seismic phases. The new model has potential applications in seismology and ultrasonic microstructure characterization.
ARTICLE | doi:10.20944/preprints201907.0334.v1
Subject: Engineering, Mechanical Engineering Keywords: polycrystals; ultrasonics; scattering; characterization; microstructure; Earth's internal core
Online: 29 July 2019 (10:35:51 CEST)
This work is a natural extension of the author’s previous work: “Multiple scattering theory for heterogeneous elastic continua with strong property fluctuation: theoretical fundamentals and applications” (arXiv:1706.09137 [physics.geo-ph]), which established the foundation for developing multiple scattering model for heterogeneous elastic continua with either weak or strong fluctuations in mass density and elastic stiffness. Polycrystalline material is another type of heterogeneous materials that widely exists in nature and extensively used in industry. In this work, the corresponding multiple scattering theory for polycrystalline materials with randomly oriented anisotropic crystallites is developed. To validate the theory, the theoretical results for a series of materials such as OFHC copper, 304 stainless steel, and Inconel 600 are compared to experimental measurements and the numerical results obtained using finite element simulations. Detailed analysis shows that the new theory is capable of predicting the dispersion and attenuation of polycrystals with satisfactory accuracy. The results also show the new model can give an estimate on the average grain size with a relative error equal to or less than ten percent. As applications in ultrasonic nondestructive evaluation, we calculated the dispersion and attenuation coefficient of one of the most important polycrystalline materials in aeronautics engineering: high-temperature titanium alloys. The effects of grain symmetry, grain size, and alloying elements on the dispersion and attenuation behaviors are examined. Key information is obtained which has significant implications for quantitatively evaluating the average grain size, monitoring the phase transition, and even estimating gradual change in chemical composition of titanium components in gas turbine engines. For applications in seismology, the velocities and Q-factors for both hexagonal and cubic polycrystalline iron models for the Earth’s uppermost inner core are obtained in the whole frequency range. Using the realistic material parameters of iron under the high temperature and high-pressure conditions calculated from ab initio simulations, the numerical results show that the Q-factors range from 0.001 to 0.05, which shows good agreement with that inferred from real seismic data. The new model predicts the velocity of longitudinal waves varies between ± 1% to ± 5 % relative to the Voight average velocity, while the velocity of transverse waves varies from ± 10% to ± 20%, which gives promising explanation to the abnormally slow transverse velocity observed in practical measurements. The numerical results support the conjecture that the Earth’s uppermost inner core is a solid polycrystalline medium. The comprehensive numerical examples show the new model is capable of capturing the most important scattering features of both ultrasonic and seismic waves with satisfactory accuracy. This work provides a universal, quantitative model for characterization of a large variety of polycrystalline materials. It also can be extended to incorporate more complicated microstructures, including ellipsoidal grains with or without textures, and even multi-phase polycrystalline materials. The new model demonstrates great potential of applications in ultrasonic nondestructive evaluation and inspection of aerospace and aeronautic structures. It also provides a theoretical framework for quantitative seismic data explanation and inversion for the material composition and structural formations of the Earth’s inner core.
ARTICLE | doi:10.20944/preprints201907.0317.v1
Subject: Earth Sciences, Geophysics Keywords: scattering; elastic waves; porous materials; dispersion; attenuation; Mohorovičić discontinuity; seismology
Online: 28 July 2019 (15:18:25 CEST)
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.