On the non-uniform motion of dislocations: The retarded elastic fields, the retarded dislocation tensor potentials and the Li\'enard-Wiechert tensor potentials

TL;DR

This paper develops a fundamental theory for the non-uniform motion of dislocations in elastic media, deriving new equations and exact solutions for elastic fields and potentials, including retarded and Li9nard-Wiechert tensor potentials.

Contribution

It introduces new fundamental formulae and exact solutions for dislocation dynamics, extending elastodynamic theory to non-uniform and arbitrary dislocation motions.

Findings

Derived retarded elastic fields and tensor potentials for dislocations.

Obtained exact closed-form solutions for dislocation loops in subsonic motion.

Analyzed singularities of elastic fields near accelerating dislocations.

Abstract

The purpose of this paper is the fundamental theory of the non-uniform motion of dislocations in two and three space-dimensions. We investigate the non-uniform motion of an arbitrary distribution of dislocations, a dislocation loop and straight dislocations in infinite media using the theory of incompatible elastodynamics. The equations of motion are derived for non-uniformly moving dislocations. The retarded elastic fields produced by a distribution of dislocations and the retarded dislocation tensor potentials are determined. New fundamental key-formulae for the dynamics of dislocations are derived (Jefimenko type and Heaviside-Feynman type equations of dislocations). In addition, exact closed-form solutions of the elastic fields produced by a dislocation loop are calculated as retarded line integral expressions for subsonic motion. The fields of the elastic velocity and elastic distortion surrounding the arbitrarily moving dislocation loop are given explicitly in terms of the so-called three-dimensional elastodynamic Li\'enard-Wiechert tensor potentials. The two-dimensional elastodynamic Li\'enard-Wiechert tensor potentials and the near-field approximation of the elastic fields for straight dislocations are calculated. The singularities of the near-fields of accelerating screw and edge dislocations are determined.