Civil Engineering Reference
In-Depth Information
and,
$
σ
33
=
(P
−
2
N)
∂
2
(ϕ
1
+
ϕ
2
)
∂
2
(ϕ
1
+
ϕ
2
)
+
∂x
1
∂x
3
Q
∂
2
(ϕ
1
+
2
N
∂
2
(ϕ
1
+
ϕ
2
)
∂
2
(ϕ
1
ϕ
2
)
ϕ
2
)
∂
2
ψ
2
∂x
1
∂x
3
+
+
+
+
+
∂x
1
∂x
3
∂x
3
σ
13
=
N
2
∂
2
(ϕ
1
+
%
ϕ
2
)
∂x
1
∂x
3
∂
2
ψ
2
∂
2
ψ
2
∂x
3
+
∂x
1
−
R
∂
2
(ϕ
1
Q
∂
2
(ϕ
1
+
ϕ
2
)
∂
2
(ϕ
1
ϕ
2
)
ϕ
2
)
∂
2
(ϕ
1
+
ϕ
2
)
+
+
σ
33
=
+
+
+
∂x
1
∂x
3
∂x
1
∂x
3
(11.28)
where
N
is the shear modulus of the material, and
P,Q
,and
R
are the elastic coefficients
of Biot defined in Chapter 6.
If
M
and
M
are set close to the forward and the backward face of the layer, respec-
tively, a matrix [
T
p
] which depends on the thickness
h
and the physical properties of
the material relates
V
p
(M)
to
V
p
(M
)
:
V
p
(M)
=
[
T
p
]
V
p
(M
)
(11.29)
The matrix elements
T
ij
have been calculated by Depollier (1989) in the following
way. Let
A
be the column vector
A
1
),(A
1
−
A
1
),(A
2
+
A
2
),(A
2
−
A
2
),(A
3
+
A
3
),(A
3
−
A
3
)
]
T
A
=
[
(A
1
+
(11.30)
and let [
(x
3
)
] be the matrix connecting
V
p
(M)
at
x
3
to
A
:
V
p
(M)
=
[
(
0
)
]
A
,
V
p
(M
)
=
[
(h)
]
A
(11.31)
The vectors
V
p
(M)
and
V
p
(M
)
are related by
V
p
(M)
=
[
(
0
)
][
(h)
]
−
1
V
p
(M
)
(11.32)
and so [
T
p
] is equal to
[
T
p
]
[
(
0
)
][
(h)
]
−
1
=
(11.33)
In order to avoid a matrix inversion, the origin of the
x
3
axis can be changed, and
the following equation can be used:
[
T
p
]
h)
][
(
0
)
]
−
1
=
[
(
−
(11.34)
Matrix [
(
0
)
]
−
1
can be evaluated analytically.
