On the non-uniform motion of dislocations: the retarded elastic fields, the retarded dislocation tensor potentials and the Liénard–Wiechert tensor potentials

Abstract

The topic 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 Liénard–Wiechert tensor potentials. The two-dimensional elastodynamic Liénard–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.

Keywords:

• dislocation dynamics
• non-uniform motion
• dislocation loop
• elastodynamics
• radiation
• retarded fields
• near-fields
• singularities

Acknowledgements

The author gratefully acknowledges the grants obtained from the Deutsche Forschungsgemeinschaft (Grant Nos. La1974/2-1, La1974/3-1). The author wishes to express his gratitude to Prof. H.O.K. Kirchner for stimulating discussions, criticism and useful remarks on an earlier version of the paper.

Notes

Notes

1. We use the usual notation β _ij,k  ≔ ∂ _k β _ij and .

2. In the literature, the notation of the dislocation density tensor and the dislocation current tensor is not unique: α (Lazar 30) =  α (Kossecka 16,41) =  α ^T(Kossecka and deWit 17) =  α ^T(Kröner 42) =  α ^T(deWit 43) = − α (Teodosiu 36) = − α ^T(Kosevich 28 and I (Lazar 30) =  I (Kossecka 16) = − I ^T(Kossecka and deWit 17) = − I (Teodosiu 36) =  I ^T(Kosevich 28).

3. In electrodynamics, the Jefimenko formulae for the electric field strength E and the magnetic field strength B are given by 23,50:

where ρ is the electric charge density, J denotes the electric current density vector, c denotes the speed of light and ε₀ is the permittivity of the vacuum.

4. The original Liénard–Wiechert potentials (scalar potential φ and vector potential A ) of a point charge read 50,55:

where q is the electric charge. Here t _c denotes the retarded time with respect to the velocity of light. They fulfil the Lorentz gauge condition: .

5. In electrodynamics, the Heaviside–Feynman formulae for the electric field strength E and the magnetic field strength B of a non-uniformly moving point charge are of the form 23,51:

6. In electrodynamics, the retarded electromagnetic potentials were originally introduced by Lorenz 78 and they read 23,50:

where ρ is the electric charge density and J denotes the electric current density vector. The idea of a retarded scalar potential was first developed by Lorenz 79 in 1861 when studying waves in the theory of elasticity. The retarded potentials fulfil the Lorentz gauge condition: (see, e.g., 49,50).