Biomedical Engineering Reference
In-Depth Information
M
3
K
m
f
a
c
o
d
ij
þ
a
c
I
F
ij
þ
a
cd
M
ij
¼
M
d
ij
II
F
iq
F
qj
g
(9.22)
and
!
!
2
a
c
1
þ M
ð
3
K
m
a
o
Þ
Mð
3
K
m
a
o
Þa
I
ð
3
K
m
a
c
2
Z
ijkm
¼
d
ij
d
km
þ
ðF
ij
d
km
þ d
ij
F
km
Þ
2
2
ð
3
K
m
Þ
Þ
!
!
F
ij
F
km
2
a
c
3
M
ð
3
K
m
a
c
o
Þa
cd
II
ðd
ij
F
kq
F
qm
þ d
km
F
iq
F
qj
Þþ b
c
1
þ M
ð
a
cd
Þ
I
þ
2
2
ð
3
K
m
ð
3
K
m
Þ
Þ
!
!
F
is
F
sj
F
kq
F
qm
2
b
c
2
þ M
a
c
I
a
cd
ðF
ij
F
kq
F
qm
þ F
km
F
iq
F
qj
Þþ b
c
3
þ M
ð
a
cd
Þ
II
ð
3
K
m
II
þ
2
2
ð
3
K
m
Þ
Þ
þc
c
1
ðd
ki
d
mj
þ d
mi
d
kj
Þþc
c
2
ðF
ki
d
mj
þ F
kj
d
mi
þ F
im
d
kj
þ F
mj
d
ki
Þ
þc
c
3
ðF
ir
F
rk
d
mj
þ F
kr
F
rj
d
mi
þ F
ir
F
rm
d
kj
þ F
mr
F
rj
d
ik
Þ:
(9.23)
Problem
9.3.1. Calculate tr
A
2
, where the Biot effective stress coefficient tensor
A
is given
by (
9.20
).
9.3.2. Derive the formulas (
9.22
) and (
9.23
) relating the Biot's parameters
M
ij
and
Z
ijkm
to fabric.
9.4 Plane Waves
The propagation of a harmonic plane wave is represented kinematically by a
direction of propagation, denoted by
n
which is a unit normal to the wave front,
and
a
or
b
, which are the directions of displacement for the wave fronts associated
with
u
and
w
, respectively. These two harmonic plane waves are represented by
h
i
h
i
n
x
v
n
x
v
u
ð
x
;
t
Þ¼
a
exp i
o
t
;
w
ð
x
;
t
Þ¼
b
exp i
o
t
;
(9.24)
where
v
is the wave velocity in the direction
n
,
x
is the position vector,
o
is the
frequency and
t
is time. The slowness vector
s
is defined as
s
(1/
v
)
n
, and the
wave speed
v
may be complex. As in elastic solid wave propagation (see Example
6.3.4), a transverse wave is characterized by
a
¼
n
¼
0, a longitudinal wave by
a
1. Substituting the representations (
9.24
) for the plane waves into the field
equations (
9.16
) and (
9.17
) one obtains equations that are in Biot (
1962a
,
b
) and
Sharma (
2005
,
2008
) and many other places,
n
¼
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