Environmental Engineering Reference
In-Depth Information
13.3.9 Plane Stress Loading
The change in the total stress in the
z-
direction is assumed
to be negligible for plane stress conditions (i.e.,
dσ
z
=
considered as constants for a small increment of stress or
strain.
The compressibility form of the water phase constitutive
equation for an unsaturated soil under three-dimensional
loading can be written as:
0).
It should be noted that plane stress loading conditions are
not common in soil mechanics applications. The soil struc-
ture constitutive relationship for the plane stress loading is
computed by setting
dσ
z
equal to zero in Eq. 13.26:
V
0
=
m
1
d
σ
mean
−
u
a
+
m
2
d
u
a
−
u
w
(13.54)
dV
w
2
1
d
σ
ave
−
2
u
a
H
d
u
a
−
u
w
(13.50)
The constitutive equation for the water phase under plane
stress loading is obtained from Eq. 13.29 by setting
dσ
z
equal to zero:
−
2
μ
E
3
3
where:
dε
v
=
+
m
1
=
coefficient of water volume change with respect to
net normal stress (i.e., 3
/E
w
) and
m
2
=
coefficient of water volume change with respect to
matric suction (i.e., 1
/H
w
).
d
u
a
−
u
w
H
w
E
w
d
σ
ave
−
2
u
a
dV
w
V
0
=
2
3
+
(13.51)
Similar constitutive equations in a compressibility form
can be written for specific loading conditions. Table 13.1
presents the
m
1
,
m
2
,
m
1
, and
m
2
coefficients for the loading
conditions described in the previous section. These coeffi-
cients are regarded as another form of the soil volumetric
deformation coefficients.
13.4 COMPRESSIBILITY FORM FOR
UNSATURATED SOIL CONSTITUTIVE RELATIONS
The constitutive relations for an unsaturated soil presented in
the preceding sections were formulated using linear elasticity
form equations. These constitutive equations can be rewritten
in a compressibility form more common to soil mechanics.
The compressibility form of the constitutive equation for the
soil structure of a saturated soil is written as
dε
v
=
m
v
d
σ
−
u
w
13.4.1 Volume-Mass Form (Soil Mechanics
Terminology)
The volume-mass properties of a soil can also be used in
the formulation of constitutive equations for an unsaturated
soil. Common soil mechanics terminology makes use of void
ratio, gravimetric water content, and degree of saturation to
define volume-mass properties of unsaturated soils.
Changes in void ratio take on the form of a deformation
state variable for a
saturated
soil giving rise to the following
constitutive equation:
de
=
a
v
d
σ
−
u
w
(13.52)
where:
m
v
=
coefficient of volume change.
(13.55)
The compressibility form for the soil structure constitu-
tive equation for an unsaturated soil under general, three-
dimensional loading is as follows:
dε
v
=
m
1
d
σ
mean
−
u
a
+
m
2
d
u
a
−
u
w
where:
a
v
=
coefficient of compressibility.
(13.53)
Void ratio and gravimetric water content can be used as
the deformation state variables for the soil structure and
water phase, respectively, for an unsaturated soil. Using
soil mechanics terminology, the change in void ratio,
de
,
of an
unsaturated
soil under three-dimensional loading can
be written as
de
=
a
t
d
σ
mean
−
u
a
+
a
m
d
u
a
−
u
w
where:
m
1
=
coefficient of volume change with respect to net
normal stress [i.e., 3
(
1
−
2
μ) /E
] and
m
2
=
coefficient of volume change with respect to matric
suction (i.e., 3
/H
).
(13.56)
The
m
1
and
m
2
coefficients of volume change in Eq. 13.53
are essentially a ratio between changes in volumetric strain
and stress state variables and can be called “compressibility
moduli.” The negative signs for the coefficients of volume
change,
m
1
and
m
2
, are the result of the elasticity-type mod-
ulus,
E
and
H
, being negative for typical soils. The
m
1
and
m
2
coefficients vary in a nonlinear manner but can be
where:
a
t
=
coefficient of compressibility with respect
to a
change in net normal stress,
d
σ
mean
−
u
a
, and
a
m
=
coefficient of compressibility with respect
to a
change in matric suction,
d
u
a
−
u
w
.
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