Geoscience Reference
In-Depth Information
264
Multiscale Geomechanics
2) Local tectonics and geology have imposed an initial anisotropic distribution that
additional stresses induced by the construction of the structure will not change. We can
imagine a summing up of the plastic mechanisms along three directions in space to be
chosen based on the direction of anisotropy. This is the idea that was implemented in the
multi-mechanism version of Hujeux's model [HUJ 85], in which three perpendicular
directions were adopted. In this model, the mechanisms of deformation are written in
terms of
σ
∆∆
. This decomposition into three deviatoric mechanisms has also been used
by other authors to simulate three-dimensional boundary value problems [KAB 91,
PRE 90]. Let
i
,
j
and
k
be three perpendicular directions composing an orthonormal
basis with unit vectors
e
k
with
k ∈ [1, 2, 3]
. The projection matrix
O
k
for a plane of
normal
e
k
can be written:
O
k
= e
(1+mod(k,3))
⊗ e
(1+mod(k+1,2))
+ e
(1+mod(k+1,3))
⊗ e
(1+mod(k,2))
[9.49]
k ∈ [1, 2, 3]
We, therefore, have:
O
1
= e
2
⊗ e
3
+ e
3
⊗ e
2
[9.50]
O
2
= e
1
⊗ e
3
+ e
3
⊗ e
1
[9.51]
O
3
= e
2
⊗ e
1
+ e
1
⊗ e
2
[9.52]
We consider that the basic deviatoric plastic strain mechanisms, for which the
hypothesisofplanestrainisvalid,aregovernedbythedeviatoricstresstensorassociated
with each projected tensor. If we look at the tensor constraints corresponding to the
proposed plane of index
k,σ
kk
, that for the sake of lightness, we note
σ
k
, we will have
(without summation on indexes):
σ
k
= σ
ii
e
i
⊗ e
i
+ σ
jj
e
j
⊗ e
j
+ σ
ij
(e
i
⊗ e
j
+ e
j
⊗ e
i
)
[9.53]
Therefore, for each mechanism we can define a mean stress
p
k
, a mean strain
ε
k
, the
devoiatoric stresses
s
k
and the deviatoric strains
ε
k
as well as their second invariants
q
k
and
k
.
3) The medium's memory of its geological history is short and it is its current
stress state which creates the anisotropy. The three planes of projection correspond to
the principal directions.
4) Frequently, surfaces appear along which displacement discontinuities can exist.
These surfaces of discontinuity occur either at the interface between two materials or
in the same material (such as faults or areas where deformation is localized on a thin
strip). It is, therefore, very important that the volumetric constitutive law be compatible
with that of the interface. Besides, experimental results show similarities between the
volumetric behavior and that of interfaces. Dilatancy dependence on the average stress
is a good example of this resemblance. The constitutive law of the interface is identical
to the model with oriented criterion, except that instead of the strain vector components
ε
n
and
γ
, we use the displacement jump vector and its normal (
[u
n
]
) and tangential
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