Diffusion of elastic waves in a continuum solid with a random array of pinned dislocations

TL;DR

This paper models the diffusion of incoherent elastic waves in a solid with randomly distributed pinned dislocations, deriving explicit formulas for the diffusion coefficient considering resonant scattering and wave velocities.

Contribution

It introduces a continuum mechanics framework with a Bethe-Salpeter equation for elastic wave diffusion in dislocation-rich solids, including resonant effects and explicit diffusion coefficient formulas.

Findings

Diffusive behavior of elastic waves is confirmed in dislocation media.

Explicit diffusion coefficient formula depending on wave velocities and dislocation length scale.

Resonant frequency effects significantly influence wave scattering and diffusion.

Abstract

The propagation of incoherent elastic energy in a three-dimensional solid due to the scattering by many, randomly placed and oriented, pinned dislocation segments, is considered in a continuum mechanics framework. The scattering mechanism is that of an elastic string of length L that re-radiates as a response to an incoming wave. The scatterers are thus not static but have their own dynamics. A Bethe-Salpeter (BS) equation is established, and a Ward-Takahashi Identity (WTI) is demonstrated. The BS equation is written as a spectral problem that, using the WTI, is solved in the diffusive limit. To leading order a diffusion behavior indeed results, and an explicit formula for the diffusion coeffcient is obtained. It can be evaluated in an Independent Scattering Approximation (ISA) in the absence of intrinsic damping. It depends not only on the bare longitudinal and transverse wave velocities but also on the renormalized velocities, as well as attenuation coeffcients, of the coherent waves. The influence of the length scale given by L, and of the resonant behavior for frequencies near the resonance frequency of the strings, can be explicitly identified. A Kubo representation for the diffusion constant can be identified. Previous generic results, obtained with an energy transfer formalism, are recovered when the number of dislocations per unit volume is small. This includes the equipartition of diffusive energy density which, however, does not hold in general. The formalism bears a number of similarities with the behavior of electromagnetic waves in a medium with a random distribution of dielectric scatterers; the elastic interaction, however, is momentum dependent.