Biomedical Engineering Reference
In-Depth Information
by
1
3
K
m
f
a
c
0
n
a
c
I
F
a
cd
II
F
2
C
¼ð
M
f
n
þ
n
þ
n
gÞ
n
:
(9.34)
The micro-macro velocity average tensor
J
is related to the fabric by
j
3
F
2
J
¼
Q
ð
j
1
1
þ
j
2
F
þ
Þ;
(9.35)
where
Q
represents a rotation matrix associated with the transformation between
the principal axes of
F
and the reference coordinate system used for
J
and
j
1
,
j
2
, and
j
3
are functions of
f
, II and III. Similarly, The flow-resistivity tensor
R
, is related to
the fabric by
r
3
F
2
R
¼
r
1
1
þ
r
2
F
þ
;
(9.36)
where
r
1
,
r
2
, and
r
3
are functions of
, II and III, and
R
is equivalent to the inverse
of the second-rank intrinsic permeability tensor
K
.
f
Problems
9.5.1. Express the direct notation equation (
9.32
) for
Q
in the indicial notation.
9.5.2. Express the equation (
9.32
) for
Q
in the matrix notation in the principal
coordinate system of the fabric tensor.
9.6 Propagation of Waves in a Principal Direction
of Material Symmetry
In this section the solution is developed for waves that propagate in the direction of
a principal axis of material symmetry. The direction of propagation is selected to be
the one direction, thus
n
are given by the vector
n
[1, 0, 0]
T
. The solution to the
problem is the solution of the 6 by 6 system of equations given by (
9.23
), thus the
values of the tensors
J
,
R
,
C
, and
Q
in the coordinate system of the principal axes of
material symmetry and at the vector
n
¼
[1, 0, 0]
T
are determined first. Under these
conditions
J
is determined from (
9.35
) with
Q
¼
¼
1
,
R
from (
9.36
),
C
from (
9.34
)
and
Q
from (
9.32
) are given by
2
3
2
3
2
3
J
11
00
0
J
22
0
00
J
33
R
11
00
0
R
22
0
00
R
33
C
11
00
000
000
4
5
;
4
5
;
4
5
J
¼
R
¼
C
¼
(9.37)
and
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