Now we use the pile from Fig. 2(a), i.e. 
with 
,
and examine the influence of one removed particle on the stress
distribution. Here we remove the third, fifth, and seventh particle
denoted with R = 3, 5, and 7 respectively, from the right in row M=7.
We relax the pile and plot the
vertical normalized stress in row M=1, i.e. at the bottom,
in Fig. 5(a). Evidently, the formerly constant stress of the
complete pile (solid line) is disturbed. We observe that the
stress decreases in the region below the missing particle at
X = 0.55, 0.65, and 0.75
for R = 7, 5, and 3 respectively. Interestingly, the stress
is minimal when following lines parallel to the slopes of the
pile, towards row M=1, starting from the vacancy.
Note that following the slopes means here: following a line
in the diamond contact network. The lines of contact are here
tilted by from the horizontal and thus are parallel to the
slopes. Going from the minimum value outwards
we observe a sudden jump to the maximum value of V(1).
When a particle close to the center of the pile is removed, i.e. R
= 7, for 
,
the stress pattern is almost symmetric to the center X=0.5, whereas the
pattern gets more and more asymmetric with decreasing R.
When a particle is removed, this particle can not longer transfer the stresses to its lower neighbors. Therefore the minimum stress is found when following the slopes starting from the missing particle. The stress which has not been carried by the missing particle has thus to be transferred along its right and left neighbors, what leads to the maximum stresses just outwards from the minimum stresses.
Figure 5:
(a) Vertical stress V(1) in row M=1 vs. X for a
pile with bumpy bottom and (solid line).
V(1) is given for piles where particle R = 7, 5, or 3,
is removed from row 7; here R counts from the right.
(b) The vertical stress V(M) is plotted for different
rows M = 1, 4, 7, 10, 13, for the pile where particle 3
is removed from row 7. Note the missing symbol for M=7
(circles).
(c) Contact network for the situation where particle R=7 is
removed from row M=7.
(d) Principal axis of stress for the situation where particle R=7 is
removed from row M=7.
In order to clarify this result we plot the vertical stresses inside the pile at different heights M = 1, 4, 7, 10, and 13 in Fig. 5(b). With increasing M, i.e. increasing height in the pile, S decreases since the weight of the part of the pile above M decreases. Inside the pile, the stress is minimal when following the slopes downward, starting from the missing particle. Interestingly, we observe an asymmetric stress also for M > 7. In Fig. 5(c) we plot the contact network for the case where particle R=7 is removed from row M=7. We observe an increase of the number of contacts only for the neighbors of the vacancy. From the principal axis of stress, in Fig. 5(d), we observe an arch-like structure, i.e. the stress below the vacancy is comparatively small. Furthermore, the direction of the major principal axis is almost vertical below the vacancy, and tilted outwards for the particles which carry larger stresses.
We learn by removing one particle from the pile, that stress
decreases below the vacancy; however, the minimum of stress
is observed when following the internal structure of the pile
downwards, i.e. lines tilted by from the horizontal. Much larger
stresses are observed outwards from the minima in stress, i.e.
an arch-like structure is found already for one missing particle.
Note that this simulation is not in contradiction to the discussion, concerning point source terms, in Ref.[16]. Wittmer et al. discuss the change an infinitesimally small mass element has onto the stress distribution and conclude that the (small) weight is propagated along ``rays'' mainly into the direction of gravity. In our case the mass removed is quite large and thus the contact network is deformed what leads to the different effects described above.