Civil Engineering Reference
In-Depth Information
The displacement potentials can be used to express the spatial derivatives of the
x
3
and the
x
1
components of
u
f
and
u
s
in Equations (11.39) - (11.41)
∂
2
∂x
1
∂
2
∂x
3
∇
·
u
s
(ϕ
1
+
ϕ
2
)
+
(ϕ
1
+
ϕ
2
)
=
(11.42)
∂
2
∂x
1
∂
2
∂x
3
∇
·
u
f
(µ
1
ϕ
1
+
µ
2
ϕ
2
)
+
(µ
1
ϕ
1
+
µ
2
ϕ
2
)
=
(11.43)
∂u
s
3
∂x
3
=
∂
2
∂x
3
∂
2
ψ
2
∂x
1
∂x
3
(ϕ
1
+
ϕ
2
)
+
(11.44)
∂
2
ψ
2
∂x
1
−
∂u
s
1
∂x
3
+
∂u
s
3
∂x
1
=
2
∂
2
(ϕ
1
+
ϕ
2
)
∂x
1
∂x
3
∂
2
ψ
2
∂x
3
+
(11.45)
and Equations (11.39) - (11.41) can be rewritten as
2
k
t
i
σ
13
=
A
i
)
sin
k
i
3
−
A
i
)
cos
k
i
3
x
3
}
N
k
i
3
{
j(A
i
+
(A
i
−
=
1
,
2
(11.46)
(k
33
−
k
t
)
[
(A
3
+
A
3
)
cos
k
33
x
3
−
A
3
)
sin
k
33
x
3
]
+
j(A
3
−
σ
33
=
i
1
,
2
{
[
−
(P
+
Qµ
i
)(k
t
+
k
i
3
)
+
2
Nk
t
]
(A
i
+
A
i
)
cos
k
i
3
x
3
+
j
[
(P
+
Qµ
i
)(k
t
+
k
i
3
)
−
2
Nk
t
]
(A
i
−
A
i
)
sin
k
i
3
x
3
}
+
2
jNk
t
k
33
(A
3
+
A
3
)
sin
k
33
x
3
−
2
Nk
t
k
33
(A
3
−
A
3
)
cos
k
33
x
3
=
(11.47)
σ
33
=
Rµ
i
)(k
t
+
k
i
3
)
A
i
)
cos
k
i
3
x
3
(Q
+
{−
(A
i
−
(11.48)
i
=
1
,
2
A
i
)
sin
k
33
x
3
+
j(A
i
−
}
A
1
),(A
2
±
A
2
),
and
(A
3
±
A
3
)
, in Equations
The coefficients of the terms
(A
1
±
elements
ij
(x
3
)
as
(11.37),
(11.38)
and
(11.46) - (11.48)
are
the
matrix
given
in
Table 11.1. In this table,
D
i
and
E
i
are given by
Qµ
i
)(k
t
+
k
i
3
)
2
Nk
t
D
i
=
(P
+
−
i
=
1
,
2
(11.49)
Q)(k
t
+
k
i
3
)
E
i
=
(Rµ
i
+
i
=
1
,
2
(11.50)
The matrix [
T
p
]
[
(
0
)
][
(h)
]
−
1
=
is given in Appendix 11.A.
11.3.4 Rigid and limp frame limits
The rigid frame limit depicts the dynamic behaviour of the material when its frame
is supposed motionless. This simplification, presented in Chapter 5, can be used for
frequencies higher than the decoupling frequency,
F
d
φ
2
/(
2
πρ
1
)
. In this frequency
domain, the visco-inertial coupling between the solid and the fluid phase is sufficiently
=
σ
×
