Environmental Engineering Reference
In-Depth Information
where
L
T
is a differential operator, σ the total stress vector, ρ density of the two-
phase mixture and
g
the acceleration of gravity;
- the incremental poro-elastoplastic constitutive relationship, which, accounting
for the definition of the Terzaghi effective stress, takes the form;
d
σ
=
d
ε −
m
α
dp
[5.2]
T
where
D
T
is the tangent material matrix, ε the strain vector, α the Biot coefficient,
m
T
= [1, 1, 1, 0, 0, 0] and
p
is the water pressure;
- the sum of the mass balance equations of water and solid skeleton;
[
]
0
⎡
⎤
n
α
−
n
∂
p
∂
ε
w
+
+
α
m
-
div
k
grad
p
+
ρ
g
=
⎢
⎣
⎥
⎦
[5.3]
K
K
∂
t
∂
t
w
s
where
n
is the porosity,
K
s
and
K
w
the solid and water compressibility modulus
respectively,
k
the permeability tensor, and ρ
w
the water density.
The tangential stiffness matrix depends on the effective stress level and the total
deformation of the solid skeleton. The permeability matrix may also depend on solid
deformation.
For the spatial discretization of equations [5.1] and [5.3] we have chosen the
finite element method, which is a multipurpose approximation following the
Galerkin approach [ZIE 00]. This procedure, accounting for equation [5.2], results
in the following system of differential equations in the time domain:
0
0
u
⎡
−
K
Q
u
⎩
⎨
⎧
F
⎭
⎬
⎫
⎡
⎤
d
⎩
⎨
⎧
⎭
⎬
⎫
⎤
⎩
⎨
⎧
⎭
⎬
⎫
u
+
=
⎢
⎣
⎥
⎦
⎢
⎣
⎥
⎦
[5.4]
T
Q
S
dt
p
0
H
p
F
w
where
K
is the global stiffness matrix,
Q
the coupling matrix,
S
the compressibility
matrix,
H
the
pe
rmeability matrix, and
F
u
and
F
w
the generalized load vectors.
Vectors
u
and
p
contain the unknowns, i.e. the nodal values of the field variables.
For temporal discretization we used a finite difference scheme, so that implicit
and explicit schemes may be used, depending on the analyst's choice. The final
matrix system is:
( )
( )
( )
⎡
θ
K
−
θ
Q
⎤
⎩
⎨
⎧
u
⎭
⎬
⎫
⎡
θ
−
T
K
1
−
θ
Q
⎤
⎩
⎨
⎧
u
⎭
⎬
⎫
=
⎢
⎣
⎥
⎦
⎢
⎣
⎥
⎦
T
Q
S
+
∆
t
θ
H
p
Q
S
−
1
−
θ
∆
t
H
p
j
+
θ
j
+
1
j
+
θ
j
