Environmental Engineering Reference
In-Depth Information
p
=
net mean stress,
13.6.6.1 Equation for Water Content
Constitutive Surface
The water content of the soil at a suction
ψ
and a net mean
stress
p
have a uniform yield stress
p
0
(i.e., consolidated
at saturation) corresponding to the pores filled with water
at a suction
ψ
s
and zero net mean stress minus the change
in volume of a group of pores under a stress state equal to
p
+
ψ
and a yield stress (i.e., at saturation)
p
0
(Fig. 13.42).
When the soil is loaded to its yield stress
p
0
(i.e., at satu-
rated condition) and then dried under a constant net mean
stress
p,
pores are subject to higher effective stresses (i.e.,
σ
−
u
w
in the water-filled state) than when drying under zero
net mean stress. Changes in the pore volume consist of two
parts: (i) irreversible volume change (i.e., plastic or collapsi-
ble volume change) and (ii) reversible volume change (i.e.,
elastic volume change). An equation for the water content
constitutive surface can be written as follows:
p
0
=
pre consolidated stress, and
w
r
=
residual water content of the soil.
13.6.6.2 Equation for Void Ratio Constitutive Surface
The volume of voids in a soil at any stress state is equal to
the volume of water plus the volume of the dry pores. The
void ratio constitutive surface can be written as follows:
e(ψ, p, p
0
)
ξ
s
G
s
w
c
(
10
x
)C
c
e
sat
−
C
c
x
−
=
e
sat
−
G
s
w
(ψ
s
,
0
,
0
)
−
w
r
G
s
dx
0
w
(ψ,p,p
0
)G
s
+
[log
(P (ψ, p, p
0
))
]
−
ξ
s
G
s
w
c
(ψ
s
)(C
c
−
C
s
)
e
sat
−
C
c
ξ
s
−
+
w
r
G
s
ξ
s
w
(ψ,p,p
0
)
=
w
(ψ
s
,
0
,
0
)
G
s
w
c
(
10
x
)(C
c
−
C
s
)
e
sat
−
C
c
x
−
d
[log
(P (
10
x
,p,p
0
)
−
x
]
dx
−
dx
w
r
G
s
w
c
(ψ
s
)
{
(C
c
−
C
s
)
log[
P(ψ,p,p
0
)
]
−
ξ
s
C
c
+
C
s
log
(ψ
+
p)
}
(e
sat
−
C
c
ξ
s
−
0
[log
(ψ
+
p)
−
ψ
s
]
G
s
w
c
(ψ
s
)C
s
e
sat
−
C
c
ξ
s
−
−
w
r
G
s
)
+
w
r
G
s
(13.129)
where:
−
C
s
log
(
1
+
p)
ξ
s
G
s
w
c
(
10
x
)C
s
e
sat
−
C
c
x
−
d
[log
(
10
x
w
(ψ
s
,
0
,
0
)
=
water content at soil suction
ψ
s
on the
initial drying curve of the slurry soil,
+
p)
−
x
]
dx
−
dx
w
r
G
s
0
[tan
−
1
(ψ
+
p
−
p
0
)
+
1
.
571]
−
(C
c
−
C
s
)
log[
P(
1
,p,p
0
)
]
(13.130)
(ψ
+
p
−
p
0
)
3
.
142
P(ψ,p,p
0
)
=
+
p
0
,
where:
ξ
s
=
log
(ψ
s
)
,
⎛
⎝
η
{
(C
c
−
C
s
)
log
P(ψ,p,p
0
)
+
C
s
log
(ψ
aev
+
p)
−
C
c
log
(ψ
aev
)
}
3
e
sat
−
C
c
log
(ψ
aev
)
−
⎞
⎠
w
(ψ
s
,
0
,
0
)
=
water content at soil suction
ψ
s
on the
initial drying curve of the slurry soil,
[tan
−
1
(ψ
+
p
−
p
0
)
+
1
.
571]
(ψ
+
p
−
p
0
)
3
.
142
w
r
G
s
ψ
s
=
1
−
ψ,
P(ψ,p,p
0
)
=
+
p
0
ξ
s
=
log(
ψ
s
),
⎛
⎝
⎞
⎠
η
{
(C
c
−
C
s
)
log[
P(ψ,p,p
0
)
]
+
C
s
log
(ψ
aev
+
p)
−
C
c
log
(ψ
aev
)
}
3
e
sat
−
C
c
log
(ψ
aev
)
−
e
sat
=
void ratio at
the reference stress state,
1 kPa,
η
=
parameter representing the stress history
of the soil,
w
c
(ψ)
=
w
sat
−
w
r
G
s
ψ
s
=
1
−
ψ
,
C
c
G
s
log
(ψ)
w
r
a
ψ
b
−
+
a
,
e
sat
=
void ratio at the reference stress state,
a, b
=
curve-fitting parameters for the soil that is
initially dried from a slurry,
η
=
parameter representing the stress history
of the soil,
w
sat
−
w
r
a
ψ
b
G
s
=
specific gravity of the soil particles,
C
c
G
s
w
sat
=
gravimetric water content at the reference
stress state,
w
c
(ψ)
=
log
(ψ)
−
+
a
,
C
c
,C
s
=
compression indices of the soil under sat-
urated conditions (i.e., zero soil suction),
a, b
=
curve-fitting parameters for the drying
curve of the initially slurry soil,
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