Geoscience Reference
In-Depth Information
228
Multiscale Geomechanics
700
0.5
600
0.49
500
0.48
400
0.47
300
0.46
200
0.45
100
0.44
0.43
0
950
1000
1050
1100
1150
1200
600
650
700
750
800
850
σ
1
(kPa)
p (kPa)
(a)
(b)
Figure 8.4.
Stress proportional loading path responses obtained with the octolinear
model [DAR 82] after a triaxial compression at
p
o
= 600 kPa
and
q = 640 kPa
.
R ∈ [−3.5; −2.5]
and with
R
∈ [0.8; 1.2]
it is possible to build the constitutive matrix
S
such as
⎡
⎤
⎡
⎤
dε
1
−
dε
R
−
R
dσ
1
R
dε
2
dσ
1
+ Rdσ
3
dσ
2
− R
dσ
3
⎣
⎦
= S
⎣
R
+
R
⎦
dε
3
[8.16]
R
dε
2
dε
2
Given the static condition [8.14], second-order work vanishes at the
ε
1
− ε
3
/
R − R
/Rε
2
peak. Furthermore the
S
determinant also vanishes because
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
dσ
1
0
0
0
R
+
R
dε
3
S
R
dε
2
dε
2
[8.17]
withanon-zerovector
(dσ
1
,dε
3
/R + R
/R dε
2
,dε
2
)
.Hence,itispossibletodefinea
generalized flow rule (called failure rule) according to these mixed conjugated variables
(see [8.17]). Finally, it should be noted that when
R
and
R
are considered to be
variables,
det
S
=0
is equivalent to equation [8.8] [PRU 09c].
To conclude with these aspects, it must be stated that every stress-strain state,
where
N
s
or
M
s
is not positive definite, is included inside the bifurcation domain
and, consequently, unstable loading directions exist. Furthermore, given a particular
loading path, when the second-order work vanishes, it means that a peak does exist for
properly conjugated variables and that a failure rule can be defined for these variables.
Eventually, effective failure takes place only when the variable showing a peak is
controlled. If, on the contrary, the conjugated variable is controlled, it is possible to
pass through this peak without obvious failure. Nevertheless, the loading path after the
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