Geoscience Reference
In-Depth Information
222
Multiscale Geomechanics
Failure in materials can be translated as the loss of stability in an equilibrium
state, which is manifested by the fact that, by slightly disturbing the current condition
(position or rate) of the equilibrium position, an unbounded response of the material
will be induced. Besides, we consider only loadings for which mass and inertial effects
can be omitted. Hence, flutter instabilities will be not studied. Under these conditions,
the Hill's criterion [HIL 58] permits the detection of all kinds of material instabilities,
which are localized or diffused. In the spirit of Hill, a stress strain state is unstable, if
material deformation can be pursued in a loading direction without any energy input.
Stability is translated by the following relationship:
∂u
j
∂x
i
∂u
j
∂x
i
V
δs
ij
d
dV > 0
∀ d
=0
[8.1]
where
V
denotes the body volume at time
t
,
s
ij
the components of the stress nominal
tensor (the transposed of the Piola-Kirchoff tensor) and
d(
∂u
j
∂x
i
)
the gradient of the
deformation rate. Whenever effects of configuration change can be disregarded, this
relation boils down to the following simplification:
V
dσ
ij
dε
ij
dV > 0
∀dε =0
[8.2]
with
dσ
being the incremental Cauchy stress tensor and
dε
the infinitesimal strain rate
tensor. In the first part of this chapter, this condition is analyzed at the material point
level, hence the relation effectively studied is the following,
d
2
W = dσ
ij
dε
ij
> 0
∀dε =0
[8.3]
where
d
2
W
is called second-order work.
This chapter consists of two sections. Section 8.2 is devoted to an analysis of relation
[8.3] in three-dimensional conditions: the contour of the domain in which instabilities
can arise, called the bifurcation domain, is given for Darve's models [DAR 95] in the
three-dimensional stress space. On the other hand, cones of unstable loading directions
for a stress state located inside this domain are also presented in the stress rate space
for the models mentioned above. Section 8.3 presents an application of this criterion
to a slope stability case. The Petacciato landslide, which occurred in March 1996 in
Italy, under a slope of only
6
◦
, is modeled. Simulation is done with the finite elements
Lagaminecode[LAG 07]developedatLiègeUniversity.Inadditiontoanon-associated
elastoplastic model for soils, this code benefits from a coupled hydromechanical model,
making a simplified description of unsaturated soils possible. With the aid of this model,
we can shed useful light on certain triggering mechanisms due to the heavy rainfall
Search WWH ::
Custom Search