Diffusion of elastic waves in a two dimensional continuum with a random distribution of screw dislocations

Abstract

We study the diffusion of anti-plane elastic waves in a two dimensional continuum by many, randomly placed, screw dislocations. Building on a previously developed theory for coherent prop-

Coherent propagation and incoherent diffusion of elastic waves in a two dimensional continuum with a random distribution of edge dislocations

“…For this reason we write down our calculations in some detail, relegating a number of the more cumbersome to several appendices. Also in an Appendix we compare the approach used here to obtain a WTI with the approach used in [29] for screw dislocations, and the consequences for the diffusion constant of anti-plane elastic waves traveling in a medium populated by a set of randomly located screw dislocations.…”

“…Thus, it diverges at low frequencies when the wavelength grows. Since in the present work, a diffusion behavior associated with long wavelengths and low frequencies is of our main interest, we introduce a Peierls-Nabarro restoring force as well as a viscous damping into the edge dislocation dynamics in a similar manner as for screw dislocations in two dimensions [29]. It yields, for small oscillations around the PN minimum, the following equation of motion…”

“…( 25)) of an ensemble of randomly distributed vibrating edge dislocations on the effective wave numbers K T,L of the coherently propagating in-plane elastic waves irrespectively of their types. In this regards, the problem under consideration is expected to be similar to that, which has been recently formulated and treated in [29] for a purely scalar case of screw dislocations in two dimensions. On the other side, the two-dimensional character of the problem is still held and exhibited through a tensorial structure of the mass operator, which makes the Green's tensor from Eq.…”

“…of the problem under consideration to proceed with the incoherent (diffusive) transport of the elastic waves. Indeed, we naturally adopt some aspects of a scalar version of the spectral approach, that was applied to a medium with random scatterers of a screw dislocation type (See, [29] and references therein for details), and generalize it to our 2d tensorial case with randomly distributed vibrating edge dislocations. Meanwhile, to save the space we use bold notations for 2d tensors or vectors from now on unless indices are explicitly required to avoid misunderstanding.…”

“…A natural evolution of this train of thought is to explore the properties of elastic waves within a medium filled with randomly placed dislocations, in the diffusive regime. In a previous paper we have treated the case of a two-dimensional medium with screw dislocations [29]. In this work we tackle the problem of a two-dimensional medium as well, but populated with edge dislocations.…”