Bead-by-bead
packing makes minimal connectivity
Each bead was placed at a stable point. (no friction)
Adding one bead adds 2 contacts
Number of contacts = 2N
Number of bead position
variables (x, y) = 2N
Isostatic network:
contact constraints are just sufficient to determine bead positions. 2N
contact forces are just sufficient to balance forces on particles N beads
Isostatic networks are floppy: if we rebuild the pile with walls shifted by d,we create a deformation that costs
no energy: a Òfree modeÓ where top shifts by e
s
yy
s
xx
Internal stresses
s
xx
s
yy
must balance to avoid motion of
free mode
d
s
xx
e
s
yy
+
= 0
The medium must obey this stress balance condition.
Cf. Cates Bouchaud, Claudel,
Wittmer, 1990Õs