In this subsection we will examine the difference between
the theoretical predictions for the stresses and the numerical
simulations, both in Ref. [11] for the 
pile.
The theory is based on
the assumption that the contact network is a diamond lattice.
Thus we perform different simulations with a pile with
and change the contact network by increasing or decreasing
the separation of the fixed particles in row M=0. The centers of
the particles in the lowermost row are separated by a distance
, with the c values c = 1/15, 0, -1/750, and -1/150.
In Fig. 3(a) and (c) we plot the vertical and horizontal
components of the stress tensor, and in Fig. 3(b) and (d)
we plot the contact network and the principal axis of the stress
tensor respectively.
Figure 3:
(a) Vertical stress V(1), in row M=1, vs. X for a
pile with bumpy bottom and . The immobile particles in row
M=0 are separated by a distance , i.e. are sqeezed
together for negative c or separated for positive c.
(b) The contact networks for the corresponding systems.
(c) Horizontal stress H(1), vs. X.
(d) The principal axis of the stress tensor for the
corresponding systems.
The interesting result is that the vertical stress in Fig. 3(a) has a dip for negative c values, the depth of which increases with increasing magnitude of c [11]. The horizontal stress in Fig. 3(c) is much larger for negative c as for positive c.
From Fig. 3(b) we observe that the assumption of
a perfect diamond lattice
for the contacts is true only for c = 1/15. The vertical
stress V(1) has a zig-zag structure that we relate
to the steps at the surface of a pile. For the naively
used c = 0 and also for small negative c = -1/750 we have
a contact network with regions of coordination number 4 and 6,
corresponding to the triangular or the diamond contact network.
For sqeezed bottom particles, i.e. c = -1/150, the contact
network is again a diamond lattice, but the orientation is
tilted outwards from the center.
From Fig. 3(d) we evidence arching for negative
c and no arching for positive c. Seemingly, a tilted
diamond lattice is neccessary for an arch to form in
this situation.
In Fig. 4(a) we present for the simulations
from Fig. 3 the angle 
, see Eq. 4
about which the major principal
axis is rotated from the horizontal in counterclockwise direction.
For c < 0 we observe a constant angle in the outer part [consistent
with the fixed principal axis (FPA) theory in Ref. [16]],
and a transition region in the center. We observe FPA only for
negative c when we also find arching.
In contrast, for 
we observe a slow continuous variation
of over the whole pile. In Fig. 4(b) we plot
the ratio of the principal axis 
and observe an
almost constant value in the outer region of the pile,
whereas in the inner part the ratio is strongly c dependent.
Figure:
(a) The angle of the major principal axis of the stress, ,
in row M=1 vs. X, from the simulations in
Fig. 3.
(b) The ratio of minor to major principal axis vs. i
X from the simulations in (a).
From a detailed comparison of the contact network and the stress
tensor we may correlate several facts: Firstly, the ratio of
the principal axis, s, seems to determine whether
the contact network is a triangular or a diamond structure,
the latter with one open contact.
For c=0 and for c=-1/750 we observe the triangular
contact network if s is large.
Secondly, the direction of the diamonds is correlated to 
,
i.e. we observe the tilted diamond lattice (for negative c) if
the major axis is tilted far enough from the horizontal.