  We test the elasticity of granular aggregates using increments of shear and
volume strain in a numerical simulation. We find that the increment in volume
strain is almost reversible, but the increment in shear strain is not. The
strength of this irreversibility increases as the average number of contacts
per particle (the coordination number) decreases. For increments of volume
strain, an elastic model that includes both average and fluctuating motions
between contacting particles reproduces well the numerical results over the
entire range of coordination numbers. For increments of shear strain, the
theory and simulations agree quite well for high values of the coordination
number.
