Geoscience Reference
In-Depth Information
300
Multiscale Geomechanics
In current practice, methods for obtaining the state of a geo-structural breakdown
are different from those used for determining their displacement. However, it would be
very interesting to use the same methods, models, and tools to analyze these two aspects.
These methods should not only detect the ultimate limit state of the structure but also
the displacements that a structure could be subjected to at the time of breakdown.
Through this example, we study how the finite element method is able to detect the
ultimatelimitstate.Weillustratetheimportanceoftheconstitutivelawontheevaluation
of the breakdown mechanism and deformation of the soil mass. The example involves
a flexible retaining wall with two inclined ties. Several criteria can be considered as
precursors of a breakdown:
1) In current practice, breakdown is associated with the non-compliance of the
balance between internal and external forces. In a numerical simulation by finite
element method, this results in non-compliance with the convergence criteria associated
with residues.
2) Intheareaofweakdiscontinuities,thatistosaywhenonlytherateofdeformation
is discontinuous, the Rice criterion [RIC 76], based on the analysis of bifurcation and
strain localization states:
Q
ep
· [d
n
v]=0
[9.93]
where
[.]
represents the jump. The acoustic tensor
Q
ep
is given by
Q
ep
= n · C
ep
· n
[9.94]
In these relations
v
is the velocity,
C
ep
is the elastoplastic tensor, and
n
is the
unit normal vector to the plane of
d
n
v
discontinuity. This is obtained by writing the
condition in such a way that this equation gives a non-trivial solution:
det(Q
ep
)=0
[9.95]
3) If we are interested in material instability, a stable state is defined by Hill's
condition [DAR 99] given by
d
2
W = dσ : dε > 0
∀dσ,dε
[9.96]
where
dσ
and
dε
are the stress and strain increments of the material with a given history.
Therefore, instability occurs when this condition is breached.
The normalized second-order work considered by Darve and Laouafa [DAR 00] is
dσ : dε
||dσ|| : ||dε||
> 0
d
2
W
norm
=
∀dσ,dε
[9.97]
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