78
8 The sine-Gordon Equation of a Dislocation
satisfy the sine-Gordon equation. However the straight dislocation is in an unstable
position of equilibrium when q = ±(2n + 1)π. A slight deviation from these positions leads to a lattice force (see also (81)) forcing the dislocation away from its
position, instead of forcing it back into its position.
We will in the following chapters concern ourselves with some of the numerous
excitation states, in other words with states of a real lattice containing dislocations,
lying energetically above a vacuum (97). We will in other words examine the solutions
of the sine-Gordon Eq. (88) or (95) as possible line forms of dislocations in a crystal.
We will not concern ourselves with the change of present line structures as a result of
influencing forces. In order to solve this problem, we would have to incorporate the
external forces F A in the general Newtonian equation (53) into our starting Eq. (79).
This way, we could take for example load tensions into consideration that leads to a
deformation and motion of dislocations by an additive tension term on the right-hand
side of the sine-Gordon equation.
We now come once again to the inertial mass m α of a dislocation. In order to do
this, we will consider a cubic crystal block with the edge lengths L made up of N
lattice atoms of the mass m. The lattice parameter is a .
This object of the total mass M = N m with N = (L/α)
3 atoms possesses 3N
degrees of freedom of motion. One would choose, in the case of small elastic oscillations, for example the 3N Cartesian coordinates of the atoms as the variables. In the
sense of Lagrangeian mechanics, we can also use any other generalised coordinates
that are capable of describing the well-defined position of the system. The art of
using the Lagrangeian mechanics is to determine such generalised coordinates that
were adapted for the physical system and thus allow an especially simple and clear
description. We made it clear at the beginning of this chapter that there are two distinct
different forms of motion for a crystal lattice: elastic and plastic deformations. The
Cartesian coordinates of mass points do not make a difference between these forms
of motion and are therefore unsuitable for the determination of plastic deformations
in a lattice. Lagrangeian coordinates of plastic displacements are, for example, the
above-introduced coordinates of dislocation q α . The plastic displacements of a lattice are always coupled to elastic displacements. One has to realise that according
to (75), below the critical shear stress, the whole motion is purely elastic. Only the
motion of overcoming the potential barrier is classified as plastical. The distinction
of this part of the motion of a lattice as plastic is a collective phenomenon. This
makes the calculation of that part of the kinetic energy belonging to the coordinates
q α far more difficult. We can, however, present a simple estimation for this:
A dislocation of the length L moves through a crystal block, as shown in Fig. 7.10.
Here, the bottom part of the cubic crystal for example glides away. If the dislocation
moves with the velocity dq/dt across the distance L, it covers this length after the
time T ,
T =
L
dq/dt
.
8 The sine-Gordon Equation of a Dislocation
satisfy the sine-Gordon equation. However the straight dislocation is in an unstable
position of equilibrium when q = ±(2n + 1)π. A slight deviation from these positions leads to a lattice force (see also (81)) forcing the dislocation away from its
position, instead of forcing it back into its position.
We will in the following chapters concern ourselves with some of the numerous
excitation states, in other words with states of a real lattice containing dislocations,
lying energetically above a vacuum (97). We will in other words examine the solutions
of the sine-Gordon Eq. (88) or (95) as possible line forms of dislocations in a crystal.
We will not concern ourselves with the change of present line structures as a result of
influencing forces. In order to solve this problem, we would have to incorporate the
external forces F A in the general Newtonian equation (53) into our starting Eq. (79).
This way, we could take for example load tensions into consideration that leads to a
deformation and motion of dislocations by an additive tension term on the right-hand
side of the sine-Gordon equation.
We now come once again to the inertial mass m α of a dislocation. In order to do
this, we will consider a cubic crystal block with the edge lengths L made up of N
lattice atoms of the mass m. The lattice parameter is a .
This object of the total mass M = N m with N = (L/α)
3 atoms possesses 3N
degrees of freedom of motion. One would choose, in the case of small elastic oscillations, for example the 3N Cartesian coordinates of the atoms as the variables. In the
sense of Lagrangeian mechanics, we can also use any other generalised coordinates
that are capable of describing the well-defined position of the system. The art of
using the Lagrangeian mechanics is to determine such generalised coordinates that
were adapted for the physical system and thus allow an especially simple and clear
description. We made it clear at the beginning of this chapter that there are two distinct
different forms of motion for a crystal lattice: elastic and plastic deformations. The
Cartesian coordinates of mass points do not make a difference between these forms
of motion and are therefore unsuitable for the determination of plastic deformations
in a lattice. Lagrangeian coordinates of plastic displacements are, for example, the
above-introduced coordinates of dislocation q α . The plastic displacements of a lattice are always coupled to elastic displacements. One has to realise that according
to (75), below the critical shear stress, the whole motion is purely elastic. Only the
motion of overcoming the potential barrier is classified as plastical. The distinction
of this part of the motion of a lattice as plastic is a collective phenomenon. This
makes the calculation of that part of the kinetic energy belonging to the coordinates
q α far more difficult. We can, however, present a simple estimation for this:
A dislocation of the length L moves through a crystal block, as shown in Fig. 7.10.
Here, the bottom part of the cubic crystal for example glides away. If the dislocation
moves with the velocity dq/dt across the distance L, it covers this length after the
time T ,
T =
L
dq/dt
.