Environmental Engineering Reference
In-Depth Information
The total strain for the unsaturated soil is the summation
of the solutions for the linear rheological elements associated
with the stress state variables
σ
eigenvalues of
A
are uniquely determined by the soil
parameters since only
κ
w
and
η
w
appear in the coefficient
matrix
A
. The external load or stress
σ
only affects the
magnitude of the strain.
−
u
w
and
σ
−
u
a
. The total
strain
ε
(
t
) is given by
z
w
1
(t)
+
z
a
1
(t)
16.11.5 Numerical Solution for Proposed Rheological
Model for Unsaturated Soils
A numerical solution was formulated for solving the two
systems of linear differential equations using the Runge-
Kutta method. The unsaturated soil simulated using the rhe-
ological model does not have to be homogeneous. Each layer
can have its own soil parameters. A numerical solution is
preferred if the soil is divided into a large number of layers.
ε (t)
=
(16.103)
z
w
1
(
∞
)
+
z
a
1
(
∞
)
where:
z
w
(t)
=
(z
w
1
(t),...,z
wn
(t))
=
solution
for the
σ
−
u
w
portion,
z
a
(t)
=
(z
a
1
(t), . . . , z
an
(t))
=
solution
for the
σ
−
u
a
portion, and
z
w1
(
∞
)
and
z
a
1
(
∞
)
=
limits
of
z
w1
(t)
and
z
a
1
(t)
,
16.11.6 Comparison of Rheological Model
with Experimental Data
The rheological model was compared to experimental results
measured on compacted kaolin. The compacted kaolin had
a dry unit weight of 13.19 kN/m
3
, void ratio of 1.07, water
content of 34.3%, and degree of saturation of 78.9% (Fred-
lund and Rahardjo, 1993a). The soil specimen was simulated
using a 10-layer rheological model with approximate soil
parameters obtained from a consolidation test where the total
stress was increased from 101.1 to 202.2 kPa.
The water phase constitutive relation can be written as
respectively, as
time goes
to
infinity.
The pore-water pressure
u
wi
(t)
and the pore-air pressure
u
ai
(t)
in the
i
th layer are respectively given by the following
equations:
u
wi
(t)
=
σ
−
κ
w
[
z
wi
(t)
−
z
w
(i
+
1
)
(t)
]
(16.104)
u
ai
(t)
=
σ
−
κ
a
[
z
ai
(t)
−
z
a(i
+
1
)
(t)
]
(16.105)
d
σ
u
w
16.11.4 Closed-Form Solution for Rheological Model
The closed-form solution for Eq. 16.90 can be expressed as
dV
w
/V
0
d
σ
−
m
1
+
m
3
u
a
=
d
σ
u
a
(16.108)
−
−
⎡
⎤
z
1
(t)
z
2
(t)
·
·
·
z
n
(t)
where:
⎣
⎦
t
dV
w
/V
0
=
change in the volume of water in the soil spec-
imen with respect to the initial volume of the
soil specimen,
e
At
z
0
e
A(t
−
s)
bds
z(t)
=
=
+
(16.106)
0
m
1
=
coefficient of water volume change with
respect to a change in net normal stress, and
The coefficient matrix
A
of the initial-value problem given
by Eq. 16.102 is tridiagonal and satisfies the condition of
a Jordan matrix. According to the Sturm theorem (Dickson
1939),
A
has
n
real distinct eigenvalues. The
n
correspond-
ing eigenvectors form the fundamental matrix for the system
of differential equations (i.e., Eq. 16.102). Matrix theory
reveals that all the eigenvalues are negative. The solution of
Eq. 16.102 has the following form:
m
3
=
coefficient of water volume change with
respect to a change in the stress state variable
σ
−
u
a
.
The following relationships for the water phase rheologi-
cal model can be inferred:
γ
w
k
w
1
z
1
m
1
+
η
w
=
z
κ
w
=
(16.109)
m
3
n
where:
α
i
e
λ
i
t
z
1
(t)
=
α
0
+
(16.107)
i
=
1
k
w
=
coefficient of permeability with respect to water,
where:
γ
w
=
unit weight of water, and
z
=
thickness of each of the layers.
α
i
=
constant (
i
=
0
,
1
,...,n
)
Similarly, the air phase relationships are
λ
i
=
eigenvalue of
A
(
λ
i
<
0
;
i
=
1
,
2
,...,n
)
γ
a
k
a
1
z
1
m
1
+
Equation 16.107 implies that the consolidation process
is primarily governed by the eigenvalues of
A
.The
n
η
a
=
z
κ
a
=
(16.110)
m
3
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