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0,1
lâche
intermédiaire
dense
100
loose
medium
dense
loose
medium
dense
lâche
intermédiaire
dense
0,08
80
0,06
60
0,04
40
compression
compression
0,02
20
0
0
0
10
20
30
4 0
5 0
6 0
7 0
0
10
20
3 0
4 0
5 0
6 0
7 0
1
(%)
1
(%)
Figure2.6.
Criticalstateobtainedforthreeinitialdensities-numericalsimulationsof
triaxialtestonsamplesofspheres[YUN08]
2.4.Analysisofthedistributionofcontactforcesinagranularmaterial
The distribution of contact forces in granular materials is a topic widely
discussed in scientific literature. The pioneering studies of Jean Biarez focusing on
the orientation of the normal directions at contact provide a preliminary piece of
information that can be described by the proposed anisotropy ratio (see section 2.3).
Once the distribution of the orientation of the contact planes is identified, it is
necessary to add two items of information: the distribution of the force intensity and
the distribution of theirorientations in the local axesdefined at each contact. A great
number of analyses, some of them experimental based on photoelastic measurement
taken in the 1950s and, increasingly, numerical analyses since the 1980s, have
shown a distribution of the contact force intensity linked to the applied stress tensor
with maximal values in the direction of the major principal stress. Very early on,
Biarez was interested in analyzing the orientation of contact forces with respect to
the local axes defined at each contact. He showed in his thesis that this variable was
linked to the loading history applied on a particular material and could thus be
considered as a local variable able to explain the irreversible behavior of these kinds
of materials. Figure 2.7, taken from Biarez's thesis, shows how he attempted to
explain this point. Indeed, for a given stress state, let us consider one of the contact
forces showing an orientation δ with respect to the normal direction at contact. An
evolution of the deviatoric stress leads to an evolution of δ equal to δ'. If the
considered evolution of the deviatoric stress corresponds to an increase, δ' will be
positive and in the case where (δ + δ') becomes equal to the local friction angle,
contactand sliding willoccurleading to irreversible(plastic)strains.Iftheevolution
of the deviatoric stress shows a decrease, δ' is then negative and it would be far less
probable that (δ + δ') could be greater than the local friction angle at contact. Then,
for unloading of a sample of granular material, very few irreversible (plastic) strains
willoccur.
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