Entropy and Material Instability in the Quasi-static Mechanics of Granular Media

1 Starting from a maximum-entropy model of granular statics, this brief note explores a possible material instability in the form of a ”stress localization” anticipated in previous work (Goddard, 2002). After a brief review of the maximum-entropy model, it is shown that a special case allows for non-convex pressure-volume response of a kind that could lead to heterogeneous stress states in an isotropically compressed granular packing. 1 STATISTICS OF KINEMATICS AND STRESS Maximum-entropy estimates for the quasi-static mechanics of granular media date back at least to the pioneering work of Kanatani (1980) on the voidage of 2D granular packings. Recent works (Bagi, 1997; Kruyt & Rothenburg, 2002; Kruyt, 2003; Goddard, 2004a) treat both stress and infinitesimal strain, the latter pointing out the necessity of some specification of a priori probability in the relevant state space. This fact is already recognized in the work of Kanatani (1980), who refers to it as the ”density of states” (a term also employed in quantum mechanics). 1.1 Delaunay triangulation, deformation and stress Following previous works, Kanatani (1980); Satake (1992); Bagi (1996), we assign to a granular assembly a (Satake) graph, which we associate with a Delaunay triangulation. The graph consists of a network of vertices or nodes, representing particle centroids, connected by edges or ”bonds”, representing nearest-neighbor pairs. The latter correspond to real and latent mechanical contacts and define the edges of an 1 in Exadaktylos, G.E. and Vardoulakis, I. G. (eds), Bifurcations, Instabilities,Degradation in Geomechanics (Proc. 7th International Workshop, Chania, Crete, 2005), Springer Verlag, 2007, pp. 145-153. 2 J.D. Goddard elementary space-filling volumes known as Delaunay simplices. In space dimension d the Delaunay simplex represents the minimal cluster of particles for which a dvolume can be assigned, with d + 1 vertices connected d(d + 1)/2 edges, defining triangles in 2D, tetrahedra (Fig. 1) in 3D, etc. The Delaunay triangulation in any dimension d suffices to define the global stress and kinematics of a granular assembly.