Environmental Engineering Reference
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(σ
−
u
a
)
0
=
initial net normal stress,
e
(u
a
−
u
w
)
0
=
initial matric suction,
w
0
=
initial water content,
D
t
=
water content index with respect to net nor-
mal stress, and
C
m
D
m
=
water content index with respect to matric
suction.
C
s
e
0
Equations 14.88 and 14.89 have the following form:
(I)
e
=
a
+
b
log
(σ
−
u
a
)
+
c
log
(u
a
−
u
w
)
(14.90)
where:
log(
u
a
-
u
w
)
a
,
b
,
c
=
constants (i.e., fitting parameters).
Lloret and Alonso (1985) studied a number of mathe-
matical equations for the description of the volume change
constitutive surface of unsaturated soils subjected to con-
fined and isotropic compression. The equations were used
to best fit experimental results on different soils and the
optimum equations were selected on the basis of minimum
fitting errors. The following conclusions were drawn for the
void ratio and degree of saturation constitutive surfaces. For
a limited range in total stress, Lloret and Alonso (1985) sug-
gested that a suitable analytical expression for the void ratio
constitutive surface is as follows:
Figure 14.65
Void ratio constitutive surface representing unload-
ing conditions (from Ho, 1988).
three planes, namely, planes I
,
II, and III. The three planes
converge at a void ratio ordinate corresponding to nominal
values of the stress state variables [i.e., log
(σ
−
u
a
)
=
0
.
1
and log
(u
a
−
u
w
)
=
0
.
1]. Plane I is perpendicular to the void
ratio versus log
(σ
u
a
)
plane. Plane III is perpendicular to
the void ratio versus log
(u
a
−
−
u
w
)
plane. Plane II represents
a transition zone between planes I and III. Plane II inter-
sects both the void ratio versus log
(σ
−
u
a
)
plane and void
+
c
log (
u
a
−
u
w
)
e
=
a
+
b
(
σ
−
u
a
)
ratio versus log
(u
a
−
u
w
)
plane. The graphical illustration
of the approximated void ratio surface for swelling portion
is presented in Fig. 14.65. The equations describing planes
I, II, and III can be written as follows:
+
d
(
σ
−
u
a
)log(
u
a
−
u
w
)
(14.91)
A more suitable equation for the void ratio surface was
given by Lloret and Alonso (1985) if the range in stress
variation is large:
⎧
⎨
C
s
log
(σ
−
u
a
)
0
e
0
+
for plane I
σ
−
u
a
C
s
log
(σ
e
=
a
+
b
log (
σ
−
u
a
)
+
c
log (
u
a
−
u
w
)
−
u
a
)
0
e
0
+
+
d
log (
σ
−
u
a
)log(
u
a
−
u
w
)
(14.92)
σ
−
u
a
for plane II
C
m
log
(u
a
−
e
=
(14.95)
u
w
)
0
⎩
+
Either of the following two equations was suggested to
describe the degree of saturation constitutive surface:
u
a
−
u
w
C
m
log
(u
a
−
for plane III
u
w
)
0
e
0
+
e
b(u
a
−
u
w
)
e
−
b(u
a
−
u
w
)
c
u
a
−
u
w
e
−
b(u
a
−
u
w
)
u
a
)
(14.93)
−
S
=
a
−
+
d(σ
−
e
b(u
a
−
u
w
)
+
where:
1
e
−
b(u
a
−
u
w
)
c
u
a
)
e
0
=
initial void ratio,
S
=
a
−
−
+
d(σ
−
(14.94)
(σ
−
u
a
)
0
=
initial net normal stress,
(u
a
−
u
w
)
0
=
initial matric suction,
where:
C
s
=
slope of the intersection line of plane I with
the void ratio versus log
(σ
−
u
a
)
plane,
a, b, c
,
d
=
constants (i.e., fitting parameters).
C
m
=
slope of the intersection line of plane III with
the void ratio versus log
(u
a
−
Ho et al. (1992) assumed a linear relation between net
normal stress and matric suction at constant void ratio. The
constitutive curves on extreme planes of the void ratio plot
are essentially linear on the semilogarithmic scale. The void
ratio surface was approximated on a semilogarithmic plot by
u
w
)
plane,
C
s
=
slope of the intersection line of plane II with
the void ratio versus log
(σ
−
u
a
)
plane, and
C
m
=
slope of the intersection line of plane II with
the void ratio versus log
(u
a
−
u
w
)
plane.
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