296
27 The Separation of Eigen Stresses
The quantities ε
I
ik and v
I
i will be denoted here by us in this relationship as structural eigen strains, so-called incompatible eigen deformations that are completely
independent from any elastic parameters or volume forces and that are thus only
determined by the dislocations and their currents. We have thus achieved our goal,
cf. also Günther [35]:
For every value of the decomposition parameter p, an elastic displacement field s exists
according to (410), so that the structural eigen strains ε I and v can be calculated from (411)
and (412) independent of the elastic parameters of the medium.
We can then choose the parameter p, so that the signal velocity in the wave
Eqs. (411) and (412) corresponds to the signal velocity c o of the sine-Gordon equation or also with the transversal sound velocity c T , namely
p
ρ
= c
2
o ≈ c
2
T .
(414)
In this case, Eqs. (411) and (412) fulfil the same Lorentz symmetry as the sineGordon equation. It is the relativistic field equations for all internal observers moving
uniformly with respect to each other that have one and the same mathematical form.
The inertial mass of the field ε
I , v, created according to Eqs. (411) and (412), also
shows the same velocity dependency as the naked mass of a dislocation introduced
in Chap. 7 and later examined in (295). We can therefore—and this situation is well
known from the discussions on the mass of an electron—no longer distinguish the
two parts of mass from one another. Measuring the inertial mass of a dislocation only
gives the sum of the naked mass and the field mass defined by Eqs. (411) and (412).
We must assume that the consequences of a quantized theory of field Eqs. (411),
(412) do in fact correspond to those of quantum electrodynamics. This means that we
principally have to, based on these equations, expect effects that correspond to the
creation and annihilation of electron–positron pairs. It is known that electrons in the
limits of medium field strengths can only be accelerated, namely by the Lorentz force.
In an extremely high-energetic electromagnetic field, there are, however, certain
supplementary specific relativistic effects. If the field energy suffices, according to
the energy mass equivalence, then electron–positron pairs are spontaneously created.
A single electron cannot be created this way because of the charge conservation.
Quantitatively, the relationship where dislocations are concerned is completely
different. Due to the fact that dislocations have a finite extend, we are dealing with
linear, thus one-dimensional objects in comparison with the ’zero-dimensional’ point
electrons, the inertial mass of one dislocation is enormously large. This means that
a highly energetic, elastic field is needed in order to accelerate this dislocation. In
comparison with this, the energy needed to create an elementary; thus, the smallest
possible dislocation loop is minutely small. The equation of charge conservation of
electrons corresponds to Eq. (371) in Chap. 24 which stated that a dislocation could
not end inside of a volume. Instead of the electron–positron pairs, closed dislocation
loops are created. These loops can be arbitrarily small, as small as the lattice structure allows. The energetic relationship is, therefore, when compared to those in the
27 The Separation of Eigen Stresses
The quantities ε
I
ik and v
I
i will be denoted here by us in this relationship as structural eigen strains, so-called incompatible eigen deformations that are completely
independent from any elastic parameters or volume forces and that are thus only
determined by the dislocations and their currents. We have thus achieved our goal,
cf. also Günther [35]:
For every value of the decomposition parameter p, an elastic displacement field s exists
according to (410), so that the structural eigen strains ε I and v can be calculated from (411)
and (412) independent of the elastic parameters of the medium.
We can then choose the parameter p, so that the signal velocity in the wave
Eqs. (411) and (412) corresponds to the signal velocity c o of the sine-Gordon equation or also with the transversal sound velocity c T , namely
p
ρ
= c
2
o ≈ c
2
T .
(414)
In this case, Eqs. (411) and (412) fulfil the same Lorentz symmetry as the sineGordon equation. It is the relativistic field equations for all internal observers moving
uniformly with respect to each other that have one and the same mathematical form.
The inertial mass of the field ε
I , v, created according to Eqs. (411) and (412), also
shows the same velocity dependency as the naked mass of a dislocation introduced
in Chap. 7 and later examined in (295). We can therefore—and this situation is well
known from the discussions on the mass of an electron—no longer distinguish the
two parts of mass from one another. Measuring the inertial mass of a dislocation only
gives the sum of the naked mass and the field mass defined by Eqs. (411) and (412).
We must assume that the consequences of a quantized theory of field Eqs. (411),
(412) do in fact correspond to those of quantum electrodynamics. This means that we
principally have to, based on these equations, expect effects that correspond to the
creation and annihilation of electron–positron pairs. It is known that electrons in the
limits of medium field strengths can only be accelerated, namely by the Lorentz force.
In an extremely high-energetic electromagnetic field, there are, however, certain
supplementary specific relativistic effects. If the field energy suffices, according to
the energy mass equivalence, then electron–positron pairs are spontaneously created.
A single electron cannot be created this way because of the charge conservation.
Quantitatively, the relationship where dislocations are concerned is completely
different. Due to the fact that dislocations have a finite extend, we are dealing with
linear, thus one-dimensional objects in comparison with the ’zero-dimensional’ point
electrons, the inertial mass of one dislocation is enormously large. This means that
a highly energetic, elastic field is needed in order to accelerate this dislocation. In
comparison with this, the energy needed to create an elementary; thus, the smallest
possible dislocation loop is minutely small. The equation of charge conservation of
electrons corresponds to Eq. (371) in Chap. 24 which stated that a dislocation could
not end inside of a volume. Instead of the electron–positron pairs, closed dislocation
loops are created. These loops can be arbitrarily small, as small as the lattice structure allows. The energetic relationship is, therefore, when compared to those in the