Civil Engineering Reference
In-Depth Information
and poroelastic media, thus making the numerical implementation accurate and easier,
was proposed by Atalla
et al
. (2001a). Using Equations (6.A.14), (6.A.19) and (6.A.20),
Equation. (13.4) can be rewritten
σ
ij
(
u
s
)e
s
ij
(δ
u
s
)
d
−
ω
2
∂p
ρ
u
s
i
δu
s
i
d
−
∂x
i
δu
s
i
d
−
σ
ij
(
u
s
,
p
)
n
j
δu
s
i
d
S
γ
φ
1
+
p
δu
s
n
d
S
=
0
Q
R
−
φ
2
ω
R
pδp
d
φ
2
∂p
∂x
i
∂(δp)
∂x
i
−
∂(δp)
∂x
i
u
s
i
d
φ(u
n
−
u
s
n
)δp
d
S
−
γ
−
2
ρ
22
φ
1
u
s
n
δp
d
S
Q
R
−
+
=
0
(δ
u
s
,δp)
∀
(13.5)
t
ij
s
ij
where
φpδ
ij
are the components of the total stress defined in Appendix
6.A. Assuming a homogeneous medium, application of the divergence theorem to
the
σ
=
σ
−
last
surface
integrals
of
Equation
(13.5)
with
the
use
of
the
vector
identity,
∇·
(
β
b)
=∇·
b
+∇β·
b)
, lead to the following set of equations
ω
2
˜
φ
1
∂p
∂x
i
δu
s
i
d
Q
R
σ
ij
(
u
s
)e
s
ij
(δ
u
s
)
d
u
s
i
δu
s
i
d
−
ρ
˜
−
γ +
+
φ
1
+
p
δu
s
i,i
d
S
Q
R
σ
ij
(
u
s
,
p
)
n
j
δu
s
i
d
S
−
−
=
0
φ
2
ω
R
pδp
d
˜
φ
1
∂(δp)
∂x
i
u
s
i
d
Q
R
φ
2
∂p
∂x
i
∂(δp)
∂x
i
−
−
γ +
+
2
ρ
22
φ
1
+
δ
p
u
s
i,i
d
S
−
Q
R
φ(u
n
−
u
s
n
)δp
d
S
=
0
−
∀
(δ
u
s
,δp)
(13.6)
Q/ R)
αρ
0
where
α
is the dynamic tortu-
osity, and summing the two previous equations, the new expression of the (
u
s
,p
)weak
formulation reads (the displacement and pressure dependence of the stress components
are taken out to alleviate the presentation)
Noting that
γ
+
φ(
1
+
=
φ/α
and
ρ
22
=
φ
2
αρ
0
ω
2
R
pδp
d
φ
2
∂p
∂x
i
∂(δp)
∂x
i
−
[
σ
ij
δe
s
ij
−
ω
2
˜
u
s
i
δu
s
i
]d
ρ
+
α
δ
∂p
∂x
i
u
s
i
d
−
φ
1
+
δ(
p
u
s
i,i
)
d
S
Q
R
φ
−
σ
ij
n
j
δu
s
i
d
S
φ(u
n
−
u
s
n
)δp
d
S
(δu
s
i
,δp)
(13.7)
−
−
=
0
∀
