  The paper considers rotations at different scales in granular materials: the
rotations of individual particles, the rolling and rigid-rotation of particle
pairs, the rotational interactions of a particle within its cluster of
neighbors, and the rotation of material regions. Numerical, Discrete Element
Method (DEM) simulations on two- and three-dimensional (2D and 3D) assemblies
show that particle rotations are diverse, that they increase with strain until
the material begins to soften, and that they are expressed in spatial patterns,
even at small strains. The interactions of a pair of particles are a
combination of three modes: a contact deformation mode, a contact rolling mode,
and a mode of rigid pair motions. Definitions are presented for each mode,
including four different definitions of contact rolling. A rolling curl is also
defined, which describes the cumulative rolling of neighboring particles around
a central particle or sub-region. At a larger scale, material deformation and
rotation are measured within small sub-regions of material, and the material
deformation can be attributed to separate contributions of contact rolling,
contact deformation, and the rigid-rotation of particle pairs. The diversity
and extend of contact rolling were measured in 2D and 3D simulations. A
dominant rolling pattern was observed, which resembles the interactions of
rolling gears. This pattern can extend to distances of at least six particle
diameters from a central particle.
