Environmental Engineering Reference
In-Depth Information
14.6.2 Soil Structure Constitutive Relationship
The constitutive relationship for the soil structure can be
written in an incremental elasticity form using elasticity
parameters (Fredlund and Rahardjo, 1993):
The unloading constitutive relationship for the soil struc-
ture is presented graphically in the form of a constitutive
surface in Fig. 14.53a. The coefficients of volume change,
m 1 and m 2 , are slopes on the soil structure constitutive sur-
face and can be obtained by differentiating the surface with
respect to net normal stress and matric suction, respectively.
The coefficients of volume change for soil structure can be
written as a function of void ratio as follows:
1
+
μ
μ
E d(σ kk
ij =
d(σ ij
δ ij u a )
3 u a ij
E
d(u a
u w )
+
δ ij
(14.48)
H
∂ε v
∂(σ mean
1
∂e
∂(σ mean
m 1 =
u a ) =
(14.50)
+
e 0
u a )
where:
1
∂ε ν
∂(u a
1
∂e
∂(u a
m 2 =
u w ) =
(14.51)
σ ij =
components of the total stress tensor for the soil
structure,
1
+
e 0
u w )
σ kk =
σ x +
σ y +
σ z ,
where:
δ ij =
Kronecker delta,
E
=
elasticity parameter for the soil structure with
respect to a change in the net normal stress,
v =
de /( 1
+
e 0 ) ,
e 0 =
initial void ratio of the soil, and
H
=
elasticity parameter for the soil structure with
respect to a change in matric suction, and
e
=
any void ratio.
μ
=
Poisson's ratio.
14.6.3 Water Phase Constitutive Relationship
The water phase constitutive relationship can be written in
an incremental compressibility form as follows (Fredlund
and Rahardjo, 1993a):
Equation 14.48 can be used to write the equation for
incremental volumetric strain using the compressibility form
coefficients of volume change:
m 1 d(σ mean
m 2 d(u a
ν =
μ a )
+
u w )
(14.49)
dV w
V 0 =
m 1 d(σ mean
m 2 d(u a
u a )
+
u w )
(14.52)
where:
1
1
σ mean =
3 σ kk =
3 x +
σ y +
σ z ) , mean net total stress,
where:
2 μ) ] E , coefficient of volume change
with respect to a change in net normal stress,
and
m 1 =
[3 ( 1
m 1
=
coefficient of water volume change with respect to
a change in net normal stress and
m 2 =
m 2
3 / H , coefficient of volume change with respect
to a change in matric suction.
=
coefficient of water volume change with respect to
a change in matric suction.
ε v = Δ
V v
V 0
θ = Δ V w
V 0
m w
m s
m s
m w
( u a - u w )
( u a - u w )
(a)
(b)
Figure 14.53 Constitutive surfaces for (a) soil structure and water phase of unsaturated, swelling
soil.
 
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