A Network Model for Granular Statics with Impenetrability Constraints

We study quasi-static deformation of dense granular packings. In the reference configuration, a granular material is under confining stress (prestress). Then the packing is deformed by imposing external boundary conditions, which model engineering experiments such as shear and compression. The deformation is assumed to preserve the local structure of neighbors for each particle, which is a realistic assumption for highly compacted packings driven by small boundary displacements. We propose a two-dimensional network model of such deformations. The model takes into account elastic interparticle interactions and incorporates geometric impenetrability constraints. The effects of friction are neglected. In our model, a granular packing is represented by a spring-lattice network, whereby the particle centers correspond to vertices of the network, and interparticle contacts correspond to the edges. We work with general network geometries: periodicity is not assumed. For the springs, we use a quadratic elastic energy function. Combined with the linearized impenetrability constraints, this function provides a regularization of the hard-sphere potential for small displacements. When the network deforms, each spring either preserves its length (this corresponds to a solid-like contact), or expands (this represents a broken contact). Our goal is to study distribution of solid-like contacts in the energyminimizing configuration. We prove that under certain geometric conditions on the network, there are at least two non-stretched springs attached to each node, which means that every particle has at least two solid-like contacts. The result implies that a particle cannot loose contact with all of its neighbors. This eliminates micro-avalanches as a mechanism for structural weakening in small shear deformation.