Motion of packings of frictional grains.
Friction plays a key role in controlling the rheology of dense granular flows. Counting the number of constraints vs the number of variables indicates that critical coordination numbers Zc=3 (in D=2) and Zc=4 (in D=3) are special, in that states in which all contacts roll without frictional sliding are naively possible at and below these average coordination numbers. We construct an explicit example of such a state in D=2 based on a honeycomb lattice. This state has surprisingly large values for the typical angular velocities of the particles. Solving for the forces in such a state, we conclude that organized shear can exist in this state only on scales l<d/I, where d is the grain diameter and I=dgamma/sqrt[P/rhog] is the dimensionless "inertia number." Above this scale the coherent shear is destabilized by the disappearance of normal forces between a significant fraction of the grains. Moving to disordered lattices, we observe that rolling regions in such lattices are characterized by an antiferromagnetic short-range ordering of the particle rotations; the frustration of this ordering links the shearing states of the grain packing to low-energy spin-glass states on the same lattice. Random lattice states are also expected to exhibit large values for the typical angular velocities and to also have regions of coherent shear limited to be smaller than l.
Halsey
Thomas C
TC
ExxonMobil Upstream Research Co., 3120 Buffalo Speedway, Houston, Texas 77098, USA. thomas.c.halsey@exxonmobil.com