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

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

“…The physics of the present problem is much richer because the dislocations have a finite length, a precise orientation and Burgers vector, and the elastic waves have two polarizations that travel at different speeds. The algebra, however, is quite close to that of 62 , and we shall refer to this reference for the details of the computation.…”

“…We have tested the methods of this paper in a simplified setting: that of the incoherent behavior of elastic waves in a two dimensional continuum with a random distribution of screw dislocations 61 and of edge dislocations 62 . The screw case is a scalar problem that keeps the whole basic physics of the diffusion behavior of elastic waves when propagating incoherently among a maze of dislocations.…”

“…Just as the Dyson equation yields the pole structure for the averaged Green's tensor, the BS equation yields the pole structure for the intensity 41 . Using the standard formalism 41,42,45,62 , the BS equation for the elastic wave diffusion in the medium with dislocations is found to be (See Appendix A)…”

“…From Eqs. (57)(58)(59)(60)(61)(62), the following leading order expression for the singular part of the intensity, Φ sing is obtained:…”

“…( 72) is the sum of two contributions, as defined in (73) and (74). The computation, sketched in Appendix F is laborious but a fairly straightforward generalization of a similar computation carried out in two dimension for elastic waves diffusing among many edge dislocations 62 . The result is, the limit of small Im[K 2 T,L ]…”