Geoscience Reference
In-Depth Information
260
Multiscale Geomechanics
ξ(δ
p
)=0
if
δ
p
≤
hys
pseudoelastic domain
0 <ξ(δ
p
) < 1
if
hys
≤ δ
p
≤
mob
hysteretic domain
ξ(δ
p
)=1
if
δ
p
≥
mob
plastic domain
It is due to these areas that we manage to find the conventional (
G − γ
) curves
used in soil dynamics. We can simulate loads from small deformations encountered in
geophysics to those of strong earthquakes. These play an important role in monotonic
behavior. The parameter
ξ
must be identified based on experimental observations. The
relationship originally proposed by Hujeux [HUJ 85] is
r−r
hys
r
hys
− r
mob
m
p
ξ(r)=
for
r
hys
≤ r ≤ r
mob
[9.30]
In the case where
b =1
and
r =1
,the flow rule is an associated one.
c) while studying the evolution of shear stress under constant
p
, we obtain the
following relation:
q =
p
· F ·
M
(1 − r)
2
a
F
˙
p
+ p
· r ·
M
·
[9.31]
The following observations may be made:
- if we assume that
F =
constant at the beginning of loading (which is justified
as long as
p
<
hys
), the
a
parameter adjusts the initial slope of the response in the
(
p
-
q
) plane,
- relations [9.19] and [9.20] show that, in this case,
F = b · β · ε
v
[9.32]
we can see that it is possible to obtain a positive work hardening (
q
> 0)oranegative
one (
q
< 0) following the sign of
ε
v
. Given the signs of different terms of the relation
[9.31], the stress softening cannot occur unless when the material dilates (
ε
v
< 0
and
˙
F<0
),
- in the particular case where M
=
M', the null dilatancy line (characteristic line)
is the same as the critical state line. We thus encounter the properties of the Cam Clay
model. In this case, if the strains are high, we should verify that:
q =M·p
[9.33]
which implies that
F =1
and, therefore,
p
= p
c
. The material has reached its critical
stress or void ratio,
- knowing that it is very difficult to experimentally reach critical state because strain
localization can occur when the state of stress and strain is no longer homogeneous
in the sample, if we assume that
φ
pp
= ψ
, we can find the friction angle at perfect
plasticity from the state where the variation of volume changes sign (Figure 9.4).
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