Environmental Engineering Reference
In-Depth Information
dε
v
d(u
a
−
u
w
)
=
1
de
d(u
a
−
u
w
)
Lloret and Ledesma (1993) provided a framework for rep-
resenting elastoplastic stress-strain relationships for unsatu-
rated soils. The yield functions and corresponding flow rules
were proposed by Alonso et al. (1990). The general stress-
strain relationships take the following form:
m
2
=
1
+
e
0
dV
V
0
dε
v
=
m
1
d
σ
mean
−
u
a
+
m
2
d
u
a
−
u
w
σ
mean
=
σ
mean
dε
v
=
d(
σ
−
u
a
δ
)
1
3
(σ
x
+
σ
y
+
σ
z
)
(total mean stress)
D
e
dε
−
h
e
d(u
a
−
u
w
)
for
F
1
≤
0 and
F
2
≤
0
e
0
=
initial void ratio
=
D
ep
dε
−
h
ep
d(u
a
−
u
w
)
for
F
1
>
0or
F
2
>
0
(13.151)
e
=
f(σ
mean
−
u
a
,u
a
−
u
w
)
; (function of void ratio
state surface)
where:
Equations 13.149 and 13.150 are based on the assumption
that the volume change of unsaturated soils is a function of
changes in net stress and soil suction. Equation 13.149 has
two unknowns in one equation since Poisson's ratio is not
known. The value of Poisson's ratio is usually estimated
or measured in the laboratory through use of triaxial tests
with measurement of lateral strains. Elastic models are gen-
erally appropriate for the analysis of monotonic net stress
and soil suction paths. However, elastic models may not
be sufficiently accurate under nonmonotonic stress paths
because of the distinction between recoverable and irrecov-
erable strains.
F
1
=
yield function for load or suction decrease,
F
1
=
f(σ
−
u
a
,u
a
−
u
w
,G)
,
F
2
=
yield function for suction increase,
F
2
=
f(σ
−
u
a
,u
a
−
u
w
,G)
, and
D
e
,H
e
=
elastic constitutive matrices defined according
to Eqs. 13.143-13.148, and
D
ep
,H
ep
=
elastoplastic constitutive matrices and
∂F
1
∂(σ
−
u
a
)
T
∂G
1
∂(σ
−
u
a
)
1
A
−
A
cr
D
ep
D
e
D
e
D
e
=
−
∂F
1
∂(σ
−
u
a
)
T
h
ep
D
e
H
s
+
D
e
H
s
=
−
13.7.6 Elastoplastic Models
Elastoplastic models (i.e., right main branch in Fig. 13.66)
can be used to address features of soil behavior such as
yield and irrecoverable strains. Most elastoplastic models
are based on the same fundamental principles but use differ-
ent yield criteria, flow rules, and compressibility functions.
Figure 13.66 lists several yield criteria used in perfectly
plastic models. Perfectly plastic formulations for saturated
soils can be extended to unsaturated soils through use of the
generalized Hooke's law and by incorporating the effect of
soil suction into the yield criterion (Pereira, 1996).
Yield surfaces can be combined with hardening rules and
cap surfaces. Hardening rules are used in order to repro-
duce changes in the size of the yield surface (i.e., isotropic
hardening) or shifts in the yield surface position (kinematic
hardening). Cap surfaces are used in order to account for
yield that occurs at stress states below failure conditions.
Numerous models have been proposed for unsaturated
soils based on a critical state framework. Some of the early
models were proposed by Karube and Kato (1989), Alonso
et al., (1990), Wheeler and Sivakumar (1995), Cui and
Delage (1996), among others. Considerable emphasis has
been given to soils compacted under potentially collapsible
conditions. The models are based on isotropic hardening
laws and on yield surfaces that expand with increasing
soil suction. The collapse phenomenon is reproduced using
appropriate modes of expansion of the yield curves and
appropriate variations in soil compressibility for different
suctions. Ongoing research continues to contribute to the
understanding of elastoplastic models for unsaturated soils.
∂F
1
∂(u
a
−
u
w
)
∂G
1
∂(σ
−
u
a
)
1
A
−
A
cr
D
e
+
H
e
if
F
2
≤
0
H
s
=
H
e
H
ep
+
if
F
2
>
0
∂
∂ε
p
T
∂F
1
∂
∂G
1
∂(σ
−
u
a
)
A
=−
∂F
1
∂(σ
−
u
a
)
T
∂G
1
∂(σ
−
u
a
)
D
e
A
cr
=−
The flow rule
G
1
=
f(σ
−
u
a
,u
a
−
u
w
,)
and
is the
hardening parameter and
∂F
1
∂(σ
−
u
a
)
T
∂F
1
∂(σ
x
−
u
a
)
∂F
1
∂(σ
y
−
u
a
)
∂F
1
∂(σ
z
−
u
a
)
=
∂F
1
∂τ
xy
∂F
1
∂τ
xz
∂F
1
∂τ
yz
×
∂G
1
∂(σ
−
u
a
)
T
∂G
1
∂(σ
x
−
u
a
)
∂G
1
∂(σ
y
−
u
a
)
∂G
1
∂(σ
z
−
u
a
)
=
∂G
1
∂τ
xy
∂G
1
∂τ
xz
∂G
1
∂τ
yz
×
The term
is generally visualized as the preconsolidation
stress for saturated conditions. Different functions for the
variables
F
1
,F
2
,G
1
, and
G
2
and soil compressibility are
defined for various proposed models.
Equations 13.141 and 13.142 can be used to provide
generic equations for the derivation of the partial differential
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