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then possible to define a tensor characterizing the geometry of these arrays by the
following equation:
n
l
i
k
l
j
k
(
l
m
)
2
A
ij
=
1
n
,
[2.7]
k
=1
l
k
being a branch vector of a closed polygon having
n
sides and
l
m
being the mean
value of the branch vectors in all the considered medium. The matrix of this tensor
can then be diagonalized allowing two variables to be defined:
- the vector
l
a
, which is the unit vector giving the major principal direction of
this tensor and thus characterizes the general direction of the considered array
(except in the particular case of a perfectly isotropic array);
- the ratio of the eigenvalues of the matrix allows what the author calls the
“lengthening degree” to be defined.
Figure 2.12.
Discretization of a granular two-dimensional material by a
complete paving of local arrays (from [NGU 08])
Figures 2.13a and 2.13b show the distributions of the volumetric strains
computed in local arrays with respect to the orientation of these arrays
l
a
and their
degree of elongation. Two increments of strain are analyzed:
- the first one is defined in the contractancy domain of a usual biaxial test;
- the second one in the dilatancy domain of this test.
It is clear from these figures that for the increment defined in the contractancy
domain, the arrays orientated in the horizontal direction show negative strains
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