8 The sine-Gordon Equation of a Dislocation
77
or, if we fully write out the lattice parameters,
q(x, t) =
a
2π
q
2π D
aσ
x ,
2π D
aρ o
t
.
(96a)
We will preferably use Eq. (95) for the following calculations. The solutions of the
equation that interest us can thus be easier examined.
Using the sine-Gordon equation, we have discovered an equation, where the solutions give us the physically possible line forms of a dislocation free from external
forces. In the region of micro-plasticity, the transitions between these line forms
constitute the second correction in our idea of a mechanism of a plastic deformation.
In the light of the importance, the sine-Gordon equation plays in theoretical
physics; it is not uninteresting to remind ourselves that we have founded this equation
only on the axiomatic system of Newton’ian mechanics (79) and a few other wellknown approximations of continuum mechanics. The validity of the sine-Gordon
equation has, in the sense of this approximation, been sufficiently proven in order
to describe a certain class of phenomena in a solid. In Chaps. 15 and 21, we will
come to realise inside the frame of this approximation that the crystalline solid is
a model for a relativistic spacetime, and furthermore, we will see because of this
model character, what becomes of relativity when we observe processes that are not
fulfilled for the considered approximations.
We as external observers have found that the sine-Gordon equation can be used to
find the physically possible deviations of a once straight dislocation line relative to
the surrounding lattice. As long as the internal observers inside of our crystal remain
at rest relative to the crystal lattice, they use at most for these measurements different
units than an outside observer would use. Everything else remains the same. Both
parties will not make any other observations that the other party makes. A problem
arises when the internal observers move relative to the crystal lattice. We will concern
ourselves with this problem in Chaps. 10–13.
A class of trivial solutions q
o of the sine-Gordon equation describes the original
state of the crystal in the sense that the dislocation possess its minimal level of energy
inside of a crystal. Dislocations then occur as infinite, straight and stabile lines being
at rest. We will name these solutions with A. Seeger [89] vacuum solutions.
3 One
can easily confirm by examining Eq. (95) that these solutions are described by
q
o
= ± 2nπ, n = 0, 1, 2, . . . .
Vacuum solutions
(97)
We also notice that the constant functions
q o = ± (2n + 1)π, n = 0, 1, 2, . . . .
Unstable solutions
(97a)
3 Seeger uses the term Enneper equation for the sine-Gordon equation. In fact, this equation was
actually examined in 1870 in the frame of differential geometry by A. Enneper [18], so that a row
of mathematical facts about the sine-Gordon equation dates back to Enneper.
77
or, if we fully write out the lattice parameters,
q(x, t) =
a
2π
q
2π D
aσ
x ,
2π D
aρ o
t
.
(96a)
We will preferably use Eq. (95) for the following calculations. The solutions of the
equation that interest us can thus be easier examined.
Using the sine-Gordon equation, we have discovered an equation, where the solutions give us the physically possible line forms of a dislocation free from external
forces. In the region of micro-plasticity, the transitions between these line forms
constitute the second correction in our idea of a mechanism of a plastic deformation.
In the light of the importance, the sine-Gordon equation plays in theoretical
physics; it is not uninteresting to remind ourselves that we have founded this equation
only on the axiomatic system of Newton’ian mechanics (79) and a few other wellknown approximations of continuum mechanics. The validity of the sine-Gordon
equation has, in the sense of this approximation, been sufficiently proven in order
to describe a certain class of phenomena in a solid. In Chaps. 15 and 21, we will
come to realise inside the frame of this approximation that the crystalline solid is
a model for a relativistic spacetime, and furthermore, we will see because of this
model character, what becomes of relativity when we observe processes that are not
fulfilled for the considered approximations.
We as external observers have found that the sine-Gordon equation can be used to
find the physically possible deviations of a once straight dislocation line relative to
the surrounding lattice. As long as the internal observers inside of our crystal remain
at rest relative to the crystal lattice, they use at most for these measurements different
units than an outside observer would use. Everything else remains the same. Both
parties will not make any other observations that the other party makes. A problem
arises when the internal observers move relative to the crystal lattice. We will concern
ourselves with this problem in Chaps. 10–13.
A class of trivial solutions q
o of the sine-Gordon equation describes the original
state of the crystal in the sense that the dislocation possess its minimal level of energy
inside of a crystal. Dislocations then occur as infinite, straight and stabile lines being
at rest. We will name these solutions with A. Seeger [89] vacuum solutions.
3 One
can easily confirm by examining Eq. (95) that these solutions are described by
q
o
= ± 2nπ, n = 0, 1, 2, . . . .
Vacuum solutions
(97)
We also notice that the constant functions
q o = ± (2n + 1)π, n = 0, 1, 2, . . . .
Unstable solutions
(97a)
3 Seeger uses the term Enneper equation for the sine-Gordon equation. In fact, this equation was
actually examined in 1870 in the frame of differential geometry by A. Enneper [18], so that a row
of mathematical facts about the sine-Gordon equation dates back to Enneper.