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.
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
1. We use the usual notation β
ij,k
≔ ∂
k
β
ij
and .
3. In electrodynamics, the Jefimenko formulae for the electric field strength
E
and the magnetic field strength
B
are given by Citation23,Citation50:
where ρ is the electric charge density,
J
denotes the electric current density vector,
c denotes the speed of light and ε
0 is the permittivity of the vacuum.
4. The original Liénard–Wiechert potentials (scalar potential φ and vector potential
A
) of a point charge read Citation50,Citation55:
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 Citation23,Citation51:
6. In electrodynamics, the retarded electromagnetic potentials were originally introduced by Lorenz Citation78 and they read Citation23,Citation50:
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
Citation79 in 1861 when studying waves in the theory of elasticity. The retarded potentials fulfil the Lorentz gauge condition:
(see, e.g.,
Citation49,
Citation50).
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