From Particle to Material Behavior: the Paths Chartered by Jean Biarez - Multiscale Geomechanics: From Soil to Engineering Projects - page 24

branch vector joining two neighboring particles l i k ; and G nj is the fabric tensor
defined from the branch vectors joining neighboring particles from the following
formula:

H jm =< n j k n m k > .

G nj ′ H jm =δ nm with ′ [2.6]

All local variables considered above are defined with “a prime” because they are
defined for neighboring particles, which allows them to be differentiated from
similar variables often used to define the local mechanism occurring at particle
contact points. Neighboring particles are defined as particles belonging to a given
array (forming a closed loop of particles in contact). These local arrays are very easy
to conceive for two-dimensional materials, but much more difficult for three-
dimensional materials. In Figure 2.11, the particles defined from their centers A, B,
C and D constitute a closed loop. Two triangles can be defined in this quadrilateral:
triangles ABD and BCD. The different vectors that combine these four points are
what we call the branch vectors between neighboring particles defined in the
quadrilateral ABCD. All the two-dimensional materials considered can be
completely paved by polygons, such as ABCD, and it is then fairly easy to define all
the variables appearing in equation [2.5].

A

D

B

C

Figure 2.11. Definition of the branch vectors between neighboring particles

These results clearly show that the relevant local scale for analyzing strains
corresponds to the local particle arrays, which agrees with the suggestions made by
Biarez in his thesis. The studies by Nguyen [NGU 08, NGU 09] indicate the specific
influence of local texture through a numerical modeling approach based on DEM. A
sample consisting of circular particles is constructed numerically. It is then possible
to define a complete discretization of the material considering all the branch vectors
joining the particles in contact (Figure 2.12). Each local array corresponds to a
closed loop of particles in contact with no other particle inside the loop. Each local
array is characterized by a number of branch vectors forming a closed loop, these
vectors being oriented in such a way that the loop is described anticlockwise. It is