Iterative solution to bulk wave propagation in polycrystalline materials

Abstract

This article reevaluates two foundational models for bulk ultrasonic wave propagation in polycrystals. A decoupling of real and imaginary parts of the effective wave number permits a simple iterative method to obtain longitudinal and shear wave attenuation constants and phase velocity relations. The zeroth-order solution is that of Weaver [J. Mech. Phys. Solids 38, 55-86 (1990)]. Continued iteration converges to the unified theory solution of Stanke and Kino [J. Acoust. Soc. Am. 75, 665-681 (1984)]. The converged solution is valid for all frequencies. The iterative method mitigates the need to solve a nonlinear, complex-valued system of equations, which makes the models more robust and accessible to researchers. An analysis of the variation between the solutions is conducted and is shown to be proportional to the degree of inhomogeneity in the polycrystal.

Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry

“…Based on the Born approximation, one can retrieve the attenuation coefficient from the imaginary part of the perturbed wavenumber by Equations (33) and (34), and the result is:…”

“…Now we examine the real part of the perturbed wavenumber that governs the phase velocity in textured polycrystalline materials, and it has never been studied in previous studies [6,39,40,59]. From Equations (33) and (34), the real part of the complex wavenumber k(p) is explicitly rewritten as:…”

“…[5] is general for all polycrystals but the successive derivations were based on the Born approximation (applicable frequency is below the geometric region where wavelength << scatter size) and its explicit expressions were limited to cubic polycrystals without crystallographic texture. A recent study [33] also targeted on texture-free polycrystals has shown that Weaver's model [5] can be converged to the Stanke and Kino model [3] using numerical iteration. Cavlet and Margerin [32] proposed a spectral function method to model the wave attenuation and phase velocity in texture-free polycrystalline materials at all frequency regions; their development [32] is limited to orthotropic symmetry crystallites [32].…”

“…Recently, Cavlet and Margerin [34] further investigated the grain shape effect on attenuation and phase velocity using the spectral function method for texture-free cubic polycrystals. All these studies [3,5,[32][33][34] have developed second-order attenuation models, since they consider some multiple scattering events in the mean Green function [35]. Another second-order attenuation model, a far-field attenuation model [36], following Weaver's approach [5], has been developed for macroscopically isotropic triclinic polycrystals of ellipsoidal grains [36] and it is reasonably close to the Stanke and Kino model [3] and spectral function method [32].…”

Grain-Size Distribution Effects on the Attenuation of Laser-Generated Ultrasound in α-Titanium Alloy

“…The second model, by Weaver [12], derived a general solution in untextured cubic-symmetry polycrystals using the Dyson and Bethe–Salpeter equation. Under the use of the Born approximation, these two seminal models gave rise to the same solution of the attenuation coefficient [13]. They were further extended to some particular cases [14,15].…”