In this chapter we extend the model of the previous chapter in two different directions; the first and most important aim is to introduce long-range interactions with a view to obtaining properly glassy behaviour, and the second is to explore the role of grain shapes in granular compaction.

The model is based on the following picture. Consider a box of sand in the presence of gravity in the jamming limit. Adopting, as in the previous chapter, a lattice-based viewpoint, we visualise this box as being constituted of rows and columns of grains. When this box is shaken along the direction of gravity, the predominant dynamical response of the sandbox is known to be in the vertical direction – recall the on- and off-lattice computer simulation results presented in earlier chapters which show that correlations in the transverse plane (i.e. along rows of grains) are negligible compared to these. Another important aspect of the jamming limit is that grain-sized voids are typically absent. The dominant dynamical mechanism in this regime is therefore grain reorientation within each column to minimise the size of the partial voids that persist. We thus focus on a column model of grains in the jamming limit.

We now extend the concept of disorder to include the effect of grain shapes. Each ordered grain occupies one unit of space, while each disordered grain occupies 1 + ∈ units of space, with ∈ a measure of the partial void trapped by misorientation.