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⎡
⎤
⎦
=
1
2
⎡
⎤
⎦
+
1
2
⎡
⎤
dε
1
dε
2
dε
3
dσ
1
dσ
2
dσ
3
|dσ
1
|
|dσ
2
|
|dσ
3
|
N
+
+ N
−
N
+
− N
−
⎣
⎣
⎣
⎦
[8.13]
Hence, this second model has eight tensorial zones and is called Model
8L
for
being octolinear. To conclude this analysis of the second-order work criterion, we now
propose a physical interpretation of the results presented above.
8.2.2.
Physical interpretation
As we saw in section 8.2.1, the second-order work criterion induces a directional
dependence of the stability; regarding a certain tensorial zone, elliptical cones of
unstable loading directions can open up (Figure 8.1) strictly inside the zone. Hence, the
real cone, according to a given stress strain state, is the compilation of several elliptical
cones truncated in their tensorial zone (Figure 8.3).
Becauseofthisdirectionalpattern,itisworthwhileconsideringproportionalloading
paths, making it possible to follow any loading rate direction. In what follows, the
selected loading variables are the stress increments, but the same mathematical
developments can be made by using the strain increments instead (as loading variables).
We propose to study the following loading paths:
⎧
⎨
⎩
dσ
1
= constant constant ∈ R
dσ
1
+ Rdσ
3
=0 R∈R−{0}
dσ
2
− R
dσ
3
=0 R
[8.14]
∈R
Given these paths, the second-order work can be re-written with the following
conjugated variables:
−
R
dε
1
−
dε
3
R
d
2
W = dσ
1
R
dε
2
dε
3
R
+
R
[8.15]
+(dσ
1
+ Rdσ
3
)
R
dε
2
+(dσ
2
− R
dσ
3
) dε
2
After a drained triaxial compression until the deviatoric ratio
q/p =0.787
, some
loading paths for fixed
R
and
R
are simulated. Figure 8.4 displays the responses of
these paths. Classical notations are used:
p =
2
dev(σ)
with
dev
σ
= σ − pI
in three-dimensions. In Figure 8.4b, the response to the triaxial
compression is not shown in order not to clutter the graph. Then, if we choose to
control the following variables:
1
3
tr(σ)
and
q =
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