  Discrete element (DEM) simulations demonstrate that granular materials are
non-simple, meaning that the incremental stiffness of a granular assembly
depends on the gradients of the strain increment as well as on the strain
increment itself. In quasi-static simulations, two-dimensional granular
assemblies were stiffer when the imposed deformation was non-uniform than for
uniform deformation. The contacts between particles were modeled as
linear-frictional contacts with no contact moments. The results are interpreted
in the context of a higher-order micro-polar continuum, which admits the
possibility of higher-order stress and couple-stress. Although the behavior was
non-simple, no evidence was found for a couple-stress or an associated
stiffness. The experimental results apply consistently to three particle shapes
(circles, ovals, and a non-convex cluster shape), to assemblies of three sizes
(ranging from 250 to 4000 particles), and at pre-peak and post-peak strains.
