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2
2
|
|
=
|
Z
xx
(
)
+|
Z
yy
(
)
M
(
)
2
.
(2
.
36)
2
|
|
|
Z
xy
(
)
+|
Z
yx
(
)
Let us define the angle
,atwhich
M
(
) has a minimum. The condition
dM
(
)
d
2
M
(
)
=
0 with
>
0
2
d
d
results in the equation
2Re
Z
3
Z
4
tan 4
=
(2
.
37)
2
2
|
Z
4
|
−|
Z
3
|
with
2Re
Z
3
Z
4
sin 4
2
2
) cos 4
+
(
|
Z
4
|
−|
Z
3
|
<
0
,
where
Z
3
=
(
Z
xy
+
Z
yx
)
/
2 and
Z
4
=
(
Z
xx
−
Z
yy
)
/
2.
This equation has two solutions,
1
and
2
=
1
+
/
2, which differ by
/
2. Thus, we obtain two perpendicular directions,
1
and
2
, with minimal
principal diagonal,
Z
xx
(
2
). These directions are
considered as principal directions of the tensor [
Z
]. The principal values of the
tensor [
Z
] are obtained on the secondary diagonal as
1
)
,
Z
yy
(
1
) and
Z
xx
(
2
)
,
Z
yy
(
1
=
Z
xy
(
1
)
=−
Z
yx
(
2
) and
2
=−
2
).
Thus, by the rotation method we derive five parameters:
Z
yx
(
1
)
=
Z
xy
(
|
1
|
,
1
=
arg
1
,
|
2
|
,
2
=
arg
2
,
1
,
(2
.
38)
which fill only five from eight degrees of freedom possessed by tensor [
Z
]. So, the
parameter space given by the rotation method is incomplete and this may result in
ambiguity. Take for instance the tensor
Z
xx
+
Z
xy
[
Z
]
=
Z
yx
Z
yy
+
where
is an arbitrary quantity. To determine
1
and calculate
Z
xy
(
1
)
,
Z
yx
(
1
)
we use (2.37) and (1.27), but these equations contain only
Z
1
,
Z
3
,
Z
4
which do
not depend on
. Hence, we will receive the same principal directions and principal
values for tensors [
Z
] with different
.
Let us consider the 2D and axially symmetric 3D-models. Here min
M
(
0.
So, the rotation method will give a tensor with zero principal diagonal and principal
directions which coincide either with longitudinal and transverse directions or with
tangential and radial directions.
)
=
