The missing stress-geometry equation in granular media

The simplest solvable problem of stress transmission through a static granular material is when the grains are perfectly rigid and have an average coordination number of z̄=d+1. Under these conditions there exists an analysis of stress which is independent of the analysis of strain and the d equations of force balance ∇jσij(r)=gi(r) have to be supported by d(d−1)/2 equations. These equations are of purely geometric origin. A method of deriving them has been proposed in an earlier paper (Edwards and Grinev, Phys. Rev. Lett. 82 (1999) 5397). In this paper alternative derivations are discussed and the problem of the “missing equations” is posed as a geometrical puzzle which has yet to find a systematic solution as against sensible but fundamentally arbitrary approaches.