Environmental Engineering Reference
In-Depth Information
where:
Substituting Eqs. 13.94, 13.95, and 13.98 into Eq. 13.97
and simplifying give
v
p
(ψ
aev
(p), p)
=
volume of the pore at a net mean stress
p
and a soil suction equal to the air-
entry value of the pore if the soil is
dried under a net mean stress
p
and
−
η
C
c
{
log
(ψ
aev
(p)
+
p
]
log
(ψ
aev
)
}
3
e
sat
−
C
c
log
(ψ
aev
)
−
ψ
aev
ψ
aev
(p)
(13.99)
1
=
where:
ψ
aev
(p)
=
air-entry value of the pore if the soil
is dried under a constant net mean
stress
p
.
e
sat
=
void ratio at saturation.
There is an unknown air-entry value in Eq. 13.99 for a
pore,
ψ
aev
(p)
, when the soil is dried under constant net mean
stress
p
. The solution of the above equation is quite complex
(Pham, 2005) but can be approximated as follows:
The relationship between soil suction and volume changes
of a pore can be derived based on the capillary equation
relating matric suction to the diameter of a capillary tube.
The following equation is obtained when assuming that the
deformations of a pore are isotropic (i.e.,
ε
x
=
ε
y
=
ε
z
):
−
η
C
c
log
(ψ
aev
+
p)
−
log
(ψ
aev
)
3
e
sat
−
C
c
log
(ψ
aev
)
ψ
aev
ψ
aev
(p)
(13.100)
=
1
ψ
a
ψ
b
=
3
V
0
3
V
0
−
V
(13.96)
Equation 13.100 presents the relationship between the air-
entry value (or water-entry value) of a pore under zero net
mean stress and the air-entry value of the pore when the
slurry soil is loaded to a mean effective stress
p
0
and then
dried under a constant net mean stress
p
. In the case where
the net mean stress
p
is smaller than
p
0
, a similar equation
can be derived to take into account the elastic deformation
when the soil is unloaded from
p
0
to
p
+
ψ
aev
:
where:
ψ
b
=
air-entry value (or water-entry value) of the pore
before deformation (i.e., initial volume of
V
0
) and
ψ
a
=
air-entry value (or water entry value) of the pore
after deformation.
(C
c
−
C
s
)
log
(p
y
)
+
C
s
log
(ψ
aev
+
p)
The air-entry and water-entry values of a pore depend on
the smallest and largest open pore diameters, respectively,
connected to the pore. The above equation does not ade-
quately describe changes in air-entry value (or water-entry
value) under isotropic net total stress loading. Therefore, the
“pore shape parameter”
η
can be introduced to assist in bet-
ter representing the relationship between the air-entry value
of a pore that is dried under zero net mean stress and a pore
that is dried under a constant net mean stress
p
:
−
C
c
log
(ψ
aev
)
3[
e
sat
−
C
c
log
(ψ
aev
)
]
1
−
η
ψ
aev
ψ
aev
(p, p
y
)
=
(13.101)
p
y
=
p
y
(ψ
aev
)
=
P(ψ
aev
,p,p
0
)
p
0
for
ψ
aev
+
p
≤
p
0
=
(13.102)
p
+
ψ
aev
for
ψ
aev
+
p > p
0
ψ
a
ψ
b
=
3
V
0
3
V
0
−
ηV
(13.97)
where:
where:
p
0
=
yield stress prior to the drying process and
p
y
=
yield stress of the soil.
η
=
pore shape parameter depending on soil type and
stress history (
η
=
1).
Equation 13.102 is a discontinuous function; however, a
mathematical technique can be used to produce a continuous
function (Pham, 2005). The technique used to produce the
continuous function is similar to that used for the volume
change SWCC for a soil (Pham and Fredlund, 2008):
The virgin compression index of a pore can be calculated
in accordance with assumptions 3 and 5:
v
p
(
0
,
1
)
V
p
(
0
,
1
)
C
c
C
c
=
(13.98)
p
y
=
p
y
(ψ
aev
)
=
P(ψ
aev
,p,p
0
)
[tan
−
1
(ψ
aev
+
p
−
p
0
)
+
1
.
571]
(ψ
aev
+
p
−
p
0
)
3
.
142
+
p
0
(13.103)
The relationship between the air-entry value of a pore
subjected to zero yield stress and dried under zero net mean
stress and the case where the pore has experienced a yield
=
where:
V
p
(
0
,
1
)
=
total volume of all pores in the soil element
at the reference stress state and
C
c
=
virgin compression index of the soil.
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