282
26 Eigen Stresses and Dislocations
∂
∂x i
α ik ≡ α ik , i = 0
that is
div α ≡ ∇ · α = 0 .
(371)
Equation (371) has an astounding physical meaning. We integrate (371) over an
arbitrary volume V and find by applying Gauss’ theorem, as well as our Eq. (369),
0 =
V
∂
∂x i
α ik dV =
(V )
α ik d A i =
(V )
db k .
(372)
The surface integral calculated here is the sum of the Burgers vectors of all dislocations coming out of the volume V . This sum is always zero: In every arbitrary
volumes, there are just as many dislocations coming into a volume as there are
dislocations coming out of a volume. We discover:
A dislocation can never end inside of a volume.
We have not yet however arrived at our aimed for destination. Calculating the
dislocation density α from the plastic distorsions of the lattice, we have found out
the one quantity which even can be measured. The dislocation density is a state
variable formed from the plastic deformation. The elastic distorsion β is not such
a measurable quantity as we have already discussed in the last chapter. We need
a relationship between the dislocation density α and the elastic strain ε, which
we could determine from the stress measurements made according to Hooke’s law
(338a). Using a mathematical operation that was also stated by Kröner [47, 48], this
functions in the following way: We construct at Eq. (370) the curl to the right, thus
∇ × β × ∇ = α × ∇ .
Formulated properly, this gives
irs kpq β sp , rq = kpq α i p , q .
We now notice that the elastic strain ε is the symmetrical part of the elastic distorsion
β,
ε ik =
1
2
(β ik + β ki ) ,
(373)
and take the special properties (358) of the Levi- Civita symbol i jk into consideration. One only then needs to calculate that
1
2
( irs kpq β sp , rq + krs i pq β sp , rq ) = irs kpq ε sp , rq .
Therefore,
irs kpq ε sp , rq =
1
2
( i pq α kp , q + kpq α i p , q ) .
(374)
26 Eigen Stresses and Dislocations
∂
∂x i
α ik ≡ α ik , i = 0
that is
div α ≡ ∇ · α = 0 .
(371)
Equation (371) has an astounding physical meaning. We integrate (371) over an
arbitrary volume V and find by applying Gauss’ theorem, as well as our Eq. (369),
0 =
V
∂
∂x i
α ik dV =
(V )
α ik d A i =
(V )
db k .
(372)
The surface integral calculated here is the sum of the Burgers vectors of all dislocations coming out of the volume V . This sum is always zero: In every arbitrary
volumes, there are just as many dislocations coming into a volume as there are
dislocations coming out of a volume. We discover:
A dislocation can never end inside of a volume.
We have not yet however arrived at our aimed for destination. Calculating the
dislocation density α from the plastic distorsions of the lattice, we have found out
the one quantity which even can be measured. The dislocation density is a state
variable formed from the plastic deformation. The elastic distorsion β is not such
a measurable quantity as we have already discussed in the last chapter. We need
a relationship between the dislocation density α and the elastic strain ε, which
we could determine from the stress measurements made according to Hooke’s law
(338a). Using a mathematical operation that was also stated by Kröner [47, 48], this
functions in the following way: We construct at Eq. (370) the curl to the right, thus
∇ × β × ∇ = α × ∇ .
Formulated properly, this gives
irs kpq β sp , rq = kpq α i p , q .
We now notice that the elastic strain ε is the symmetrical part of the elastic distorsion
β,
ε ik =
1
2
(β ik + β ki ) ,
(373)
and take the special properties (358) of the Levi- Civita symbol i jk into consideration. One only then needs to calculate that
1
2
( irs kpq β sp , rq + krs i pq β sp , rq ) = irs kpq ε sp , rq .
Therefore,
irs kpq ε sp , rq =
1
2
( i pq α kp , q + kpq α i p , q ) .
(374)