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path and critical state: Δ
e
= 0.1. These two values have to be compatible (on the
condition that
C
d
is constant and does not depend on
p'
, which was found to be the
case). On
p'
constant stress paths, we must have relation [5.16] with η
= q
pp
/p' = M
,
where the index “
pp
” indicates the perfect plastic state:
2
η
pp
ee eC
−
=
− Δ= −
log 1
+
= −
0.3
C
[5.16]
pp
C
d
d
M
For loading with
p'
constant, starting from the NC state, Biarez adopted the
following model:
-Δ
e
= 0.1
C
d
= 1/3 = 0.33
[5.17]
which gives equation [5.18] for the isotropic loading at the NC state (ISL):
e
(ISL) =
e
0
+ 0.1 -
C
c
(π
0
- log
p'
)
[5.18]
where:
- for sands,
e
0
= e
max
, π
0
=
2,
C
c
= (e
max
- e
min
)/
1.9,
p'
in kPa;
- for clays,
e
0
= (2.7/100
) w
L
,
π
0
=
0.54,
C
c
=
0.009
(w
L
-
13
)
,
p'
in kPa.
5.4.3.
Overconsolidated state (C
s
)
Let point (
p'
ic
- e
C
) be the maximum isotropic stress state and point (
p'
i0
- e
OC
) be
the “initial” unloading stress state (see Figure 5.9).
p'
i0
can be noted
p'
i
or σ'
30
. The
point obtained (
p'
i0
- e
OC
) is easily characterized by its OCR (
p'
iC
/p'
i0
) for clays, but
with great difficulty for sands, as we have just seen above. Therefore, the (
p'
ic
- e
C
)
point is rarely observable.
Biarez [SAI 97], proposed a new parameter for overconsolidation, or
generalized
over-consolidation
,
e
NC
- e
OC
, which could process sands and clays together: instead
of the abscissae, he uses the ordinates. This new formulation has the benefit of not
only introducing the history of the loading, log (
p'
ic
/p'
i0
), but moreover the nature of
the material, via (
C
c
- C
s
). It is therefore more general than the usual formulation
and requires a model for
C
c
as well as for
C
s
.
The model of
C
c
for clays has already been given by [5.4].
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