An Expression for the Granular Elastic Energy
  Granular Solid Hydrodynamics (GSH) is a broad-ranged continual mechanical description of granular media capable of accounting for static stress distributions, yield phenomena, propagation and damping of elastic waves, the critical state, shear band, and fast dense flow. An important input of GSH is an expression for the elastic energy needed to deform the grains. The original expression, though useful and simple, has some draw-backs. Therefore, a slightly more complicated expression is proposed here that eliminates three of them: (1) The maximal angle at which an inclined layer of grains remains stable is increased from $26^\circ$ to the more realistic value of $30^\circ$. (2)Depending on direction and polarization, transverse elastic waves are known to propagate at slightly different velocities. The old expression neglects these differences, the new one successfully reproduces them. (3) Most importantly, the old expression contains only the Drucker-Prager yield surface. The new one contains in addition those named after Coulomb, Lade-Duncan and Matsuoka-Nakai -- realizing each, and interpolating between them, by shifting a single scalar parameter. 
Yimin Jiang, Hepeng Zheng, Zheng Peng, Liping Fu, Shixiong Song,   Qicheng Sun, Michael Mayer and Mario Liu
cond-mat.soft