26 Eigen Stresses and Dislocations
283
Here, the right-hand side is nothing other than ink ε, and for the left-hand side, Kröner
introduced the term η, so that we arrive from Eq. (374) at the famous well-known
Kröner equation,
ink ε = η
(374a)
with
η ik =
1
2
( i pq α kp , q + kpq α i p , q ) .
(375)
We have achieved our first goal. Using Eq. (374), it is possible to calculate the eigen
stresses that are created by a certain distribution of dislocations. As long as these
dislocations are present as a static distribution, where they do not move through the
crystal, Eqs. (374) and (375) are the only equations that we need, together with the
equations of motion (343) and the material equations, Hooke’s law (338a), in order
to calculate the stresses, including the eigen stresses. It is astounding, and these facts
will occupy our line of thought that the formulation of equation (374), having only the
eigen strain as contents, is completely free of any material parameters. For the elastic
strain caused by external forces, a corresponding formulation cannot principally be
made.
Before we include the motion of dislocations into our considerations, we will
refer the reader to another omission which we will include in our depiction. These
are the so-called momentum stresses. It is not completely correct that the real crystal
that we approximate using a continuum does not react mechanically to a rotation
of its volume elements V , as we assumed in Chap. 25. In fact, because of the
presence of dislocations a relative rotation of the volume elements occurs that also
reacts energetically. We will not take these effects into consideration and refer the
reader to Kröner’s [53] paper.
We now consider a single straight dislocation with the Burgers vector b, whose
line direction is described by us using the unit vector t. If A is an area element of
the size A, which is chosen perpendicular to the line direction t, then according
to Eq. (369), the following applies: A i α ik = A α ik = t i b k . For a single straight
dislocation on the area A in the t-direction and with the Burgers vector b we can
write
α ik =
t i b k
A
.
(376)
This dislocation may now move with the velocity V. It is evident that a motion of a
dislocation in its own line direction makes no physical sense. In fact, the line direction
must traverse across the area F during a motion. The vector n = V × t stands
perpendicular on this area, and is thus a vector n with the components n i = irs V r t s .
The unit vector n o constructed from this is
n oi =
irs V r t s
|V |
.
(377)
283
Here, the right-hand side is nothing other than ink ε, and for the left-hand side, Kröner
introduced the term η, so that we arrive from Eq. (374) at the famous well-known
Kröner equation,
ink ε = η
(374a)
with
η ik =
1
2
( i pq α kp , q + kpq α i p , q ) .
(375)
We have achieved our first goal. Using Eq. (374), it is possible to calculate the eigen
stresses that are created by a certain distribution of dislocations. As long as these
dislocations are present as a static distribution, where they do not move through the
crystal, Eqs. (374) and (375) are the only equations that we need, together with the
equations of motion (343) and the material equations, Hooke’s law (338a), in order
to calculate the stresses, including the eigen stresses. It is astounding, and these facts
will occupy our line of thought that the formulation of equation (374), having only the
eigen strain as contents, is completely free of any material parameters. For the elastic
strain caused by external forces, a corresponding formulation cannot principally be
made.
Before we include the motion of dislocations into our considerations, we will
refer the reader to another omission which we will include in our depiction. These
are the so-called momentum stresses. It is not completely correct that the real crystal
that we approximate using a continuum does not react mechanically to a rotation
of its volume elements V , as we assumed in Chap. 25. In fact, because of the
presence of dislocations a relative rotation of the volume elements occurs that also
reacts energetically. We will not take these effects into consideration and refer the
reader to Kröner’s [53] paper.
We now consider a single straight dislocation with the Burgers vector b, whose
line direction is described by us using the unit vector t. If A is an area element of
the size A, which is chosen perpendicular to the line direction t, then according
to Eq. (369), the following applies: A i α ik = A α ik = t i b k . For a single straight
dislocation on the area A in the t-direction and with the Burgers vector b we can
write
α ik =
t i b k
A
.
(376)
This dislocation may now move with the velocity V. It is evident that a motion of a
dislocation in its own line direction makes no physical sense. In fact, the line direction
must traverse across the area F during a motion. The vector n = V × t stands
perpendicular on this area, and is thus a vector n with the components n i = irs V r t s .
The unit vector n o constructed from this is
n oi =
irs V r t s
|V |
.
(377)