Environmental Engineering Reference
In-Depth Information
can be considered as the deformation state variables or a
change in fluid storage. The deformation variables can be
written as
V
w
/V
0
for the water phase and
V
a
/V
0
for the
air phase.
the strain-stress constitutive relations in the
x-, y-,
and
z-
directions can be written in the following form:
σ
x
−
u
w
E
μ
E
(σ
y
+
σ
z
−
ε
x
=
−
2
u
w
)
(13.12)
σ
y
−
u
w
E
μ
E
(σ
x
+
σ
z
−
ε
y
=
−
2
u
w
)
(13.13)
13.3 VOLUME-MASS CONSTITUTIVE
RELATIONS
σ
z
−
u
w
E
μ
E
(σ
x
+
σ
y
−
ε
z
=
−
2
u
w
)
(13.14)
Constitutive relations for an unsaturated soil can be for-
mulated by combining selected deformation state variables
with appropriate stress state variables. Stress tensors com-
prised of stress state variables for an unsaturated soil were
presented in Chapter 3. Deformation tensors comprised of
deformation state variables must satisfy continuity require-
ments. Combining deformation state variables and stress
state variables in the form of empirical equations results in
the definition of volumetric deformation properties. Several
forms of constitutive relations are discussed in the follow-
ing sections. The constitutive relations can be used to predict
volume changes (i.e., overall volume change and phase vol-
ume changes) that occur as a result of changes in the stress
state variables.
where:
σ
x
=
total normal stress in the
x-
direction,
σ
y
=
total normal stress in the
y-
direction,
σ
z
=
total normal stress in the
z-
direction,
=
E
modulus of elasticity or Young's modulus for the
soil structure, and
μ
=
Poisson's ratio.
The sum of the normal strains
ε
x
,
ε
y
, and
ε
z
constitutes the
volumetric strain
ε
v
. For a saturated soil, the overall volume
change of the soil is equal to the water volume change since
soil particles are essentially incompressible.
The constitutive relations for an unsaturated soil can be
formulated as an extension of the equations used for a sat-
urated soil while using the appropriate stress state variables
(Fredlund and Morgenstern, 1976). Let us assume that the
soil behaves as an isotropic, linear elastic material. The fol-
lowing constitutive relations are expressed in terms of the
σ
−
u
a
and
u
a
−
u
w
stress state variables. The formulation
is similar in form to that proposed by Biot in 1941. The soil
structure constitutive relations associated with the normal
strains in the
x-, y-,
and
z-
directions are as follows:
13.3.1 Elasticity Form
There are two main approaches that can be used in estab-
lishing the stress-deformation relationships. These are the
“mathematical” approach and the “semiempirical” approach.
In the mathematical approach, each component of the defor-
mation state variable tensor is expressed as a linear combi-
nation of the stress state variables or vice versa. In other
words, the relationship between the stress and deformation
state variables is expressed by a series of linear equations.
The difficulty with this approach is the large number of soil
properties that must be determined.
The semiempirical approach involves several assumptions
which are based on experimental evidence from observing
the behavior of many materials (Chou and Pagano, 1967):
(1) normal stress does not produce shear strain, (2) shear stress
does not cause normal strain, and (3) a shear stress component
τ
causes only one shear strain component,
γ
. In addition, the
principle of superposition is assumed to be applicable to cases
involving small deformations.
The semiempirical approach is generally adopted in soil
mechanics and is used herein to formulate the constitutive
relations for unsaturated soils. The constitutive equations
must be experimentally tested to ensure uniqueness. The
uniqueness theorem for an elastic solid with a positive
definite strain energy function states that there exists a
one-to-one correspondence between elastic deformations
and stresses (Fung, 1965).
The constitutive relations for the soil structure in sat-
urated soil mechanics can be formulated in the form of
a generalized Hooke's law using effective stress variables
σ
−
u
w
. For an isotropic and linearly elastic soil structure,
σ
x
−
u
a
E
μ
E
(σ
y
+
σ
z
−
u
a
−
u
w
H
ε
x
=
−
2
u
a
)
+
(13.15)
σ
y
−
u
a
E
μ
E
(σ
x
+
σ
z
−
u
a
−
u
w
H
ε
y
=
−
2
u
a
)
+
(13.16)
σ
z
−
u
a
E
μ
E
(σ
x
+
σ
y
−
u
a
−
u
w
H
ε
z
=
−
2
u
a
)
+
(13.17)
where:
H
=
modulus of elasticity for the soil structure with
respect to a change in matric suction,
u
a
−
u
w
.
The constitutive equations associated with the shear defor-
mations are
τ
xy
G
γ
xy
=
(13.18)
τ
yz
G
γ
yz
=
(13.19)
τ
zx
G
γ
zx
=
(13.20)
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