  We study quasi-static deformation of dense granular packings. The packing is
deformed by imposing external boundary conditions, which model engineering
experiments such as shear and compression. 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. When the network deforms, each spring either preserves its length
(this corresponds to a solid-like contact), or expands (this represents a
broken contact). We prove that under certain geometric conditions on the
network, in the energy-minimizing configuration there are at least two
non-stretched springs attached to each node, which means that every particle
has at least two solid-like contacts. This eliminates micro-avalanches as a
mechanism for structural weakening in small shear deformation.
