Biomedical Engineering Reference
In-Depth Information
convey their algebraic complexity. However contemporary symbolic algebraic
software solves this type of eigenvalue problem in a few keystrokes.
Problems
9.4.1. Using the indicial notation substitute the representations (
9.24
) for the plane
waves into the field equations (
9.16
) and (
9.17
) and derive the coupled
equations (
9.28
) and (
9.29
) for the velocity and amplitudes of the two
waves in the fixed direction
n
.
9.4.2. How do you use your favorite symbolic algebraic software to solve (
9.30
)in
the case where the parameters are all specified numerically.
9.5 Fabric Dependence of the Tensors Q
,
C
,
J
, and
R
The governing equations for anisotropic poroelasicity for quasi-static and dynamic
poroelasticity were developed and extended to include the dependence of the consti-
tutive relations upon a pore structure fabric tensor
F
as well as the porosity (Cowin
1985
,
2003
,
2004
; Cowin and Cardoso
2011
; Cardoso and Cowin
2011
,
2012
).
Formulas relating the acoustic tensor
Q
, the flow-resistivity tensor
R
and the tensor
C
, representing the interaction of the velocity fields
u
and
w
, to the fabric tensor
F
are
obtained in this subsection. The dependence of the elastic acoustic tensor
Q
upon the
fabric tensor
F
is obtained by substituting (
9.23
) into the first of (
9.27
),
c
c
1
þ
c
c
2
c
c
3
F
2
Q
¼ð
tr
f
F
n
n
gþ
tr
f
n
n
gÞ
1
c
c
2
F þ
þ
q
1
n n þ
q
2
ðF n n þ n n FÞ
c
c
3
F
2
F
2
F
2
þ
þ
q
3
ð
n
n
þ
n
n
Þþ
q
4
F
n
F
n
F
2
F
2
q
6
F
2
F
2
þ
q
5
ð
F
n
n
þ
n
F
n
Þþ
n
n
;
(9.32)
where
2
3
K
m
3
K
m
a
o
Þ
a
o
Þ
a
I
M
ð
M
ð
c
c
1
þ
a
c
1
þ
c
c
2
þ
a
c
2
q
1
¼
;
q
2
¼
;
2
2
3
K
m
3
K
m
ð
Þ
ð
Þ
2
3
K
m
M
ð
a
o
Þ
a
II
M
ð
a
I
Þ
c
c
3
þ
a
c
3
b
c
1
þ
q
3
¼
;
q
4
¼
2
;
3
K
m
2
3
K
m
ð
Þ
ð
Þ
2
M
a
I
a
II
ð
3
K
m
a
II
Þ
M
ð
b
c
2
þ
b
c
3
þ
q
5
¼
2
;
q
6
¼
2
:
(9.33)
ð
3
K
m
Þ
Þ
, II, and III.
The formula for the tensor
C
is obtained by substituting (
9.8
) into the second of
(
9.27
) and then employing (
9.20
), thus
The six quantities defined in (
9.33
) are scalar-valued functions of
f
1
3
K
m
f
a
c
0
n
a
c
I
F
a
cd
II
F
2
½
C
¼
M
f
n
þ
n
þ
n
g
n
:
(9.34)
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