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

TL;DR

This paper investigates how elastic waves scatter off pinned dislocation segments in a solid, using perturbation theory to sum the series and analyze frequency-dependent effects, with implications for material properties.

Contribution

It provides a formal summation of the scattering series and introduces a renormalization approach for low frequencies, connecting physical quantities to experimentally determined parameters.

Findings

Perturbation series sums to a geometric series for the mass operator.

Divergent quantities are renormalized to zero at low frequencies.

Physical quantities depend on a dislocation length cutoff and two parameters.

Abstract

The coherent propagation of elastic waves in a solid filled with a random distribution of pinned dislocation segments is studied to all orders in perturbation theory. It is shown that, within the independent scattering approximation, the perturbation series that generates the mass operator is a geometric series that can thus be formally summed. A divergent quantity is shown to be renormalizable to zero at low frequencies. At higher frequencies said quantity can be expressed in terms of a cut-off with dimensions of length, related to the dislocation length, and physical quantities can be computed in terms of two parameters, to be determined by experiment. The approach used in this problem is compared and contrasted with the scattering of de Broglie waves by delta-function potentials as described by the Schr\"odinger equation.