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2.5. Analysis of local arrays
In every study concerning the behavior of a granular material with a change of
scale approach, one of the main questions is how to define the relevant local scale.
Even to this day, we do not have a completely satisfactory answer but, in his thesis
[BIA 62], Biarez had already posed this question. Indeed part of the analyses that he
proposed was to be seen at the contact scale (particularly for defining the
geometrical anisotropy of contact presented in section 2.3); he also understood that
another interesting level of analysis could be the local arrays of particles. He
analyzed, in particular, the balance of simple arrays consisting of four particles
(Figure 2.10) and showed that, for a given stress state, the stability of these arrays
depended on their geometry.
Figure 2.10. Analysis of the equilibrium of regular arrays of discs
(figure from [BIA 62])
Recent studies have shown that it is not possible to change the scale of the
kinematical variables in granular materials by only considering the relative
displacements at contact points. It has nevertheless been well established that the
strain at the representative elementary volume can be obtained by using an
averaging technique that has to do with the relative displacement of neighboring
particles; neighboring particles being defined for a two-dimensional granular
material as particles belonging to a given local array. The change of scale formula
can then be defined as follows [CAM 00]:
N
N
δ l j k
δ l i k
1
2 N
l k n n k
l k n n k
δε ij =
G nj +
G ni
,
[2.5]
k =1
k =1
where l k is the length of the branch vector defined by means of Delaunay
discretization corresponding to neighboring particles; δ l i k is the increment of the
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