27 The Separation of Eigen Stresses
295
We finally find using (403)
v i −
ρ
p
∂
2
∂t 2 v i +
+
1
p
∂
∂t
p
−2
∂
∂x r
S ir pq C pqmn ε mn
+ 2
∂
∂x i
S rr pq C pqrs ε mn
= −J ir , r −J ri , r +J rr , i −
1
p
∂
∂t
f i .
(409)
Instead of Eqs. (389) and (390) for the isotropic continuum, Eqs. (408) and (409)
apply for the general case of Hooke’s tensor C used in the determination of the fields
ε and v for an arbitrary prescribed distribution of dislocations α and their currents
J as well as arbitrary volume forces f .
The structure of the left-hand sides of these equations is somewhat complicated
when dealing with the general case of Hooke’s tensor and its 21 independent parameters. The following decoupling of these equations can be achieved, see Günther
[35]. We introduce an elastic displacement vector s, so that the portions ε
I and v
I
can be removed from the corresponding parts of the total elastic deformation ε and
the matter velocity v according to
ε ik = ε
I
ik +
1
2
(s i , k +s k , i ) ,
v i = v
I
i +
∂
∂t
s i .
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(410)
One can now show, for this we refer to Günther [35]: The quantities ε
I
ik and v
I
i
simply fulfil the d’Alembertian wave equations
ε
I
ik −
ρ
p
∂
2
∂t 2 ε
I
ik = −
1
2
( mni α nk , m + mnk α ni , m )
−
1
2
( mni α mn , k + mnk α mn , i ) −
ρ
p
∂
∂t
1
2
(J ik + J ki )
(411)
and
v
I
i −
ρ
p
∂
2
∂t 2 v
I
i = − J ir , r − J ri , r + J rr , i ,
(412)
if the elastic vector s fulfils the equations of the classical theory of elasticity, thus
equations of the type (342a) with a certain additional force created by the strain
components ε
I
mn ,
ρ
∂
2 s i
∂t 2 =
1
2
C rimn (s m , nr +s n , mr ) +
+ f i + p
−2
∂
∂x r
S ir pq C pqmn ε mn
+ 2
∂
∂x i
S rr pq C pqrs ε mn
.
(413)