Elastic wave propagation through a random array of dislocations

Abstract

International audienceA number of unsolved issues in materials physics suggest there is a need for an improved quantitative understanding of the interaction between acoustic (more generally, elastic) waves and dislocations. In this paper we study the coherent propagation of elastic waves through a two dimensional solid filled with randomly placed dislocations, both edge and screw, in a multiple scattering formalism. Wavelengths are supposed to be large compared to a Burgers vector and dislocation density is supposed to be small, in a sense made precise in the body of the paper. Consequently, the basic mechanism for the scattering of an elastic wave by a line defect is quite simple ("fluttering"): An elastic wave will hit each individual dislocation, causing it to oscillate in response. The ensuing oscillatory motion will generate outgoing (from the dislocation position) elastic waves. When many dislocations are present, the resulting wave behavior can be quite involved because of multiple scattering. However, under some circumstances, there may exist a coherent wave propagating with an effective wave velocity, its amplitude being attenuated because of the energy scattered away from the direction of propagation. The present study concerns the determination of the coherent wavenumber of an elastic wave propagating through an elastic medium filled with randomly placed dislocations. The real part of the coherent wavenumber gives the effective wave velocity and its imaginary part gives the attenuation length (or elastic mean free path). The calculation is performed perturbatively, using a wave equation for the particle velocity with a right hand side term, valid both in two and three dimensions, that accounts for the dislocation motion when forced by an external stress. In two dimensions, the motion of a dislocation is that of a massive particle driven by the incident wave; both screw and edge dislocations are considered. The effective velocity of the coherent wave appears at first order in perturbation theory, while the attenuation length appears at second order

Multiple scattering of elastic waves by pinned dislocation segments in a continuum

“…(21). Note also that the two dimensional problem of transverse (anti-plane) waves scattered by an infinite straight screw dislocation is a scalar Schrödinger problem 19 with a potential k|V |k proportional to k 2 . The analysis we review below also holds when the point obstacle is modelled by even derivatives of the delta function 20 .…”

Effective dynamic properties of 3D composite materials containing rigid penny-shaped inclusions

“…Previous results have shown that distributed crack-like defects may cause a decrease in the phase velocity and an increase in the wave attenuation. The efficiency and the applicability ranges of 2D homogenization analysis of elastic wave propagation through a random array of scatters of different shapes and dilute concentrations based on the BEM and Foldy-type dispersion relations were demonstrated also by many authors, for instance, by Maurel et al [20], Sato and Shindo [21], Tourin et al [22]. However, to the author's knowledge, the overall average dynamic response of 3D composite material containing randomly distributed and movable penny-shaped inclusions with a much larger rigidity than that of the matrix material has not been investigated for moderate and high frequencies.…”

“…With Eq. (20), the amplitude of a plane time-harmonic elastic wave propagating in the n -direction can be expressed as…”

Numerical Simulation of Wave Propagation in 3D Elastic Composites with Rigid Disk-Shaped Inclusions of Variable Mass

“…Previous results have shown that distributed crack-like defects may cause a decrease in the phase velocity and an increase in the wave attenuation. The efficiency and the applicability ranges of 2D homogenization analysis of elastic wave propagation through a random array of scatters of different shapes and dilute concentrations based on the BEM and Foldy-type dispersion relations were demonstrated also by many authors, for instance, in the papers [13,14]. In 3D case this approach was applied for the numerical simulation of the average dynamic response of composite material containing rigid disk-shaped inclusions of equal mass only [15].…”