From Particle to Material Behavior: the Paths Chartered by Jean Biarez - Multiscale Geomechanics: From Soil to Engineering Projects - page 21

Figure 2.7. Explanatory scheme on the evolution of the orientation of a contact force
during a loading or unloading (figure taken from [BIA 62])

With the numerical discrete modeling tool (DEM), it is possible to analyze this
kind of local variable directly. It is then very convenient to construct the following
three stress tensors:

- the stress tensor computed from the local contact forces using the equation of
change of scale:

σ ij = 1
V

k 

F i k l j k

[2.2]

- the part of this tensor only linked to the normal components of the contact
forces:

σ ij N = 1
V

k 

F i Nk l j k [2.3]

- the part of this tensor only linked to the tangential components of the contact
forces:

σ i T = 1
V

k 

F i Tk l j k

[2.4]

where V is the volume in which the contact forces are considered, F k is the contact
force vector applied to contact k and  l k is the vector joining the centers of mass of
the two particles in contact at contact-point k .

If component σ 11 T is considered, it is clear that this component is positive when
components F 1 Tk and l 1 k have essentially the same signs; whenever the global
component is negative, the local components have essentially opposite signs.