An important quantity that allows insight into the state of
the system is the stress tensor 
[25, 26],
which we identify in the static case with
where the indices 
and 
indicate the
coordinates, i.e. x and z in 2D, see Fig. 1.
This stress tensor
is an average over all contacts of the particles within
volume 
, with q denoting the distance between the
center of the particle and the contact point, and f denoting
the force acting at the contact point. Throughout this study we
average over the contacts of one particles (i) to get the stresses
for one realization.
From a static configuration of ``soft'' particles we may now calculate
the components of the stress tensor 
, and 
and also define 
,
,
and 
.
Since we neglected tangential forces the particles are
torque-free and we observe only symmetric stress tensors,
i.e. 
. The eigenvalues
of are thus 
, and the major eigenvalue
is tilted by an angle
from the horizontal in counter clockwise direction.
In order to find the correct scaling for the stress we assume
like Liffman et al. [8, 11],
as a simplified example, a rigid triangle with the density 
, the
width l, the height h, and the mass 
.
Since the material is rigid, we find a constant force at the
supporting surface, so that the pressure is also constant 
. Thus we will scale the stress by the
pressure p and furtheron use the dimensionless stress
with the volume a = h l / 2 of the triangular pile.
The vertical component will be abbreviated with 
,
the horizontal component with 
, and the shear
component 
.
Besides the components of S we will also plot the stress tensor in its
principal axis representation, i.e. for each particle
we plot the scaled major principal axis along 
and the minor
axis in the perpendicular direction.
