Geoscience Reference
In-Depth Information
262
Multiscale Geomechanics
a = a
1
+ ξ(r)(a
1
− a
2
)
[9.36]
1) We note that this deviatoric mechanism alone cannot take into account the
isotropic behavior of soils. For example, an isotropic consolidation path cannot be
modeled by deviatoric mechanisms alone, unless it is in critical condition. The
experimental results show that the points representing samples at a state of isotropic
consolidation and those representing the samples at the perfect plasticity state in a
triaxial test are positioned on two parallel lines distanced
d
in the (
e − log p
) plane
and that we do not have
p
= exp
1/b
p
c
which is obtained from the yield surface
for
q =0
.This aspect is taken into account by introducing an additional isotropic
mechanism. The yield function is written as follows:
f
iso
(p
,r
iso
(ε
v
),p
c
(ε
v
)) = |p
s
|−d·r
iso
· p
c
(ε
v
) ≤ 0
[9.37]
This implies that in the (
p
− q
) plane, the axis
p
is crossed by a vertical line
whose position changes with
ε
v
. In this mechanism:
p
m
= p
, and
p
c
= p
h
with
p
h
= p
− r
iso
h
· d · p
c
[9.38]
p
d · p
c
r
iso
h
=
[9.39]
The dilatancy law is associated and, consequently, the evolution of the degree of
mobilization of the mechanism (
r
iso
)isgivenby
r
iso
=ε
v
iso
(1 − r
iso
)
2
[9.40]
c
s
(p
c
/p
ref
)
where
ε
v
iso
is the plastic volume variation due to this mechanism. Parameter
c
s
controls
the mobilization of this mechanism, depending on whether it is a monotonic (
c
m
)or
cyclic (
c
c
) one.
In the previous model, we have just summarized a way to take into account the
main characteristics of soil behavior observed in the course of oedometric and triaxial
tests. These formulations do not take into account the role of intermediate stress and
initial or induced anisotropy. To describe the three-dimensional behavior of soils, the
model should be generalized. Multimechanism plasticity is a powerful tool that can
integrate these aspects.
9.3.4.
Generalizing the simplified model
Several situations can be imagined. To better illustrate the mechanisms that govern
the behavior in each situation, we introduce a tensorial projection operator, denoted
O
∆
acting on a vector
v
as follows:
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