Environmental Engineering Reference
In-Depth Information
for specified changes in total stress, pore-air, and pore-water
pressures:
16.6.6 Coupled Consolidation Equations
for Unsaturated Soil
The derivation of the theory of coupled consolidation is
based on the assumption that the air phase is continuous.
Several other assumptions used in the derivation are similar
to those adopted by Terzaghi (1943) and Biot (1941). The
assumptions can be listed as follows: (1) material is isotropic,
(2) reversibility of stress-strain relations, (3) linearity of
stress-strain relations, (4) strains are small, (5) pore-water is
incompressible, (6) coefficients of permeability with respect
to water and air phases are functions of the volume-mass
soil properties during the consolidation process, and (7) the
effects of air diffusing through water, air dissolving in the
water phase, and the movement of water vapor are ignored.
There are five unknowns of deformation and volumetric
variables to be solved in a three-dimensional consolida-
tion problem. The variables are the displacements in the
x
-,
y
-, and
z
-directions (i.e.,
u
,
v
, and
w
, respectively)
and the water and air volume changes (i.e.,
dV
w
and
dV
a
).
Displacements in the
x
-,
y
-, and
z
-directions are used to
compute the total volume change. The five unknowns can
be obtained from three equilibrium equations and two con-
tinuity equations (i.e., the water and air phase continuities).
These equations are summarized in the following sections.
d(σ
y
−
u
a
)
dV
w
V
0
=
d(σ
x
−
u
a
)
+
E
w
E
w
d
σ
z
−
u
a
d
u
a
−
u
w
+
+
(16.57)
E
w
H
w
where:
dV
w
=
water volume change in the soil element,
E
w
=
water volumetric modulus associated with a
change in
σ
−
u
a
, and
H
w
=
water volumetric modulus associated with a
change in
u
a
−
u
w
.
Substituting Eqs. 16.54, 16.55, and 16.56 into Eq. 16.57
allows the water volume change in the soil to be written as
β
w
2
d
u
a
−
u
w
dV
w
V
0
=
β
w
1
dε
v
+
(16.58)
where
E
E
w
(
1
1
H
w
−
3
β
E
w
β
w
1
=
β
w
2
=
−
2
μ)
16.6.7 Stress Equilibrium Equations
The stress state for an unsaturated soil element should satisfy
the following equilibrium conditions:
16.6.5 Air Phase Constitutive Relations
The constitutive equation for the air phase defines the change
in volume of air in an unsaturated soil element for spec-
ified changes in the total stress, pore-air, and pore-water
pressures:
∂τ
yx
∂y
∂σ
x
∂x
∂τ
zx
∂z
+
+
=
0
(16.61)
∂τ
xy
∂x
+
∂σ
y
∂y
+
∂τ
zy
∂z
=
0
(16.62)
d(σ
y
−
u
a
)
dV
a
V
0
=
d(σ
x
−
u
a
)
+
E
a
E
a
∂τ
xz
∂x
∂τ
yz
∂y
∂σ
z
∂z
+
+
=
0
(16.63)
d(σ
z
−
u
a
)
d(u
a
−
u
w
)
+
+
(16.59)
E
a
H
a
The effect of the body force is assumed to be negligible.
Substituting Eqs. 16.54, 16.55, and 16.56 into Eqs. 16.61,
16.62, and 16.63 gives the following form for the equilib-
rium equations:
where:
dV
a
=
air volume change in the soil element,
E
a
=
air volumetric modulus associated with a change
in
σ
β
∂
u
a
−
u
w
G
∂ε
v
∂x
∂u
a
∂x
−
u
a
, and
2
u
G
∇
+
−
+
=
0
H
a
=
air volumetric modulus associated with a change
in
u
a
−
1
−
2
μ
∂x
(16.64)
u
w
.
β
∂
u
a
−
u
w
G
∂ε
v
∂y
∂u
a
∂y
Substituting Eqs. 16.54, 16.55, and 16.56 into Eq. 16.59
allows the air volume change to be written as
2
v
G
∇
+
−
+
=
0
1
−
2
μ
∂y
(16.65)
β
∂
u
a
−
u
w
β
a
2
d
u
a
−
u
w
dV
a
V
0
=
β
a
1
dε
v
+
G
∂ε
v
∂z
∂u
a
∂z
(16.60)
2
w
G
∇
+
−
+
=
0
1
−
2
μ
∂z
(16.66)
where
where
E
1
H
a
−
3
β
E
a
∂
2
∂x
2
+
∂
2
∂y
2
+
∂
2
∂z
2
β
a
1
=
β
a
2
=
2
∇
=
E
a
(
1
−
2
μ)
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