Environmental Engineering Reference
In-Depth Information
where:
τ
xy
=
shear stress on the
x-
plane in the
y-
direction (i.e.,
τ
xy
=
τ
yx
),
τ
yz
=
shear stress on the
y-
plane in the
z-
direction (i.e.,
τ
yz
=
τ
zy
),
τ
zx
=
shear stress on the
z-
plane in the
x-
direction (i.e.,
τ
zx
=
τ
xz
), and
G
=
shear modulus.
The modulus of elasticity
E
in the above equations is de-
finedwith respect to a change in the net normal stress
σ
−
u
a
during uniaxial loading. The above constitutive equations
can be applied in an incremental manner when the stress-
strain curve is nonlinear. Figure 13.7 defines the sign
convention using a typical stress-strain curve. An incremental
procedure using small increments of stress and strain can be
used to apply the linear elastic formulation to a nonlinear
stress-strain curve. The nonlinear stress-strain curve is
assumed to be linear within each stress and strain increment.
The elastic moduli
E
and
H
have negative signs as indicated
in Fig. 13.7 and may vary in magnitude from one increment
to another. The soil structure constitutive relations associated
with the normal strains can be written in an incremental form:
Figure 13.7
Definition of variables for nonlinear stress-strain
curve for soil.
d
σ
x
−
u
a
E
d
u
a
−
u
w
H
(13.21)
The above equation can be simplified as follows:
E
d
σ
y
+
σ
z
−
2
u
a
+
μ
dε
x
=
−
3
1
d
σ
mean
−
u
a
+
H
d
u
a
−
u
w
(13.26)
−
2
μ
E
3
dε
v
=
d
σ
y
−
u
a
E
d
u
a
−
u
w
H
(13.22)
E
d
σ
x
+
σ
z
−
2
u
a
+
μ
dε
y
=
−
where:
d
u
a
−
u
w
H
(13.23)
The above equations represent the general elasticity con-
stitutive relations for the soil structure. The left-hand side
of the equations refers to a change in the deformation state
variable while the right-hand side contains changes in the
stress state variables. A change in the volumetric strain of
the soil for each increment,
dε
v
, can be obtained by sum-
ming the incremental changes in normal strains in the
x-, y-,
and
z-
directions:
d
σ
z
−
u
a
E
σ
mean
=
mean total normal stress [i.e.,
(σ
x
+
σ
y
+
σ
z
)
/3].
E
d
σ
x
+
σ
y
−
2
u
a
+
μ
dε
z
=
−
The volumetric strain change
dε
v
is equal to the volume
change of the soil element divided by the initial volume of
the element:
dV
v
V
0
dε
v
=
(13.27)
The initial volume
V
0
refers to the volume of the soil
element at the start of the volume change process. There-
fore,
V
0
remains constant for all increments. The change in
volumetric strain,
dε
v
, at the end of each increment can be
computed from Eq. 13.26, and the volume change of the soil
element,
dV
v
, is obtained from Eq. 13.27. The summation
of the volumetric strain changes for each increment gives
the final volumetric strain of the soil:
dε
v
=
dε
x
+
dε
y
+
dε
z
(13.24)
where:
dε
v
=
volumetric strain change for each increment.
dε
v
ε
v
=
(13.28)
Substituting Eqs. 13.21, 13.22, and 13.23 into Eq. 13.24
gives
13.3.2 Water Phase Constitutive Relationship
The soil structure constitutive relationship is not sufficient to
completely describe volume changes in an unsaturated soil.
Either an air or water phase constitutive relationship must
be formulated. It is suggested that the water phase may be
3
1
d
σ
x
+
σ
y
+
σ
z
3
−
u
a
−
2
μ
dε
v
=
E
H
d
u
a
−
u
w
3
+
(13.25)
Search WWH ::
Custom Search