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where
p
c
= p
c0
exp
βε
v
[9.87]
As the material is normally consolidated:
p
p
c0
d =
[9.88]
which results in:
1
bβ
(
q
rMp
− 1+bln d)
ε
v
=
[9.89]
Therefore, the volume change at constant
p
depends on
b
,
d
,
β
and
K
parameters.
The hardening parameters act through variable
r
which reaches 1 at perfect plasticity.
For a given type of material (fixed
b
), the volume change with respect to
q/(
M
p
)
depends on
d
and
β
parameters which are correlated to liquidity limit
w
L
for the clays
and relative density
D
r
for the sands. For the elastic part, under constant
p
path,
K
can only change with void ratio. However, the elastic part being negligible compared
to plastic part, it does not influence
c
d
value significantly.
9.4.2.5.
Behavior domains
γ
ela
and
γ
hyd
can be obtained from test results. They have been the subject of
numerous research studies. Vucetic [VUC 94] suggests the range of 0.01 to 0.04%
for
γ
hys
. These values are based on tests performed on sands of different (
D
r
∈
[20 − 85%]
), under various confining stresses
σ
0
∈
[25-190 kPa] (see Figure 9.11).
For clays, similar results are given.
γ
hys
is about 0.1% and hardly varies with over-
consolidation.
9.4.2.6.
Unsaturated soil parameters
Given the formulation of the model for unsaturated soil, only a few parameters
depend exclusively on unsaturation. Fleureau and his co-authors [FLE 93] have shown
the validity of the correlations suggested by Biarez and Favre [BIA 72] between the
C
c
and
w
L
parameters on drainage paths applied to normally consolidated clays. Indeed,
the parameters necessary to model the mechanical behavior are:
p
desat
,
p
resat
,
π
c
(pc)
and
(π
c
)
. Zerhouni [ZER 91] suggested a relation between suction at desaturation
p
desat
and the liquidity limit (see Figure 9.12). Khalkhali and Khabbaz [KHA 98]
obtained the following relation illustrated in Figure 9.13 for
χ
, which determines the
effective stress (see relation [9.60]):
p
c
p
c
desat
)
−0.55
χ =(
[9.90]
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