The paper develops the stiffness relationship between the movements and
forces among a system of discrete interacting grains. The approach is similar
to that used in structural analysis, but the stiffness matrix of granular
material is inherently non-symmetric because of the geometrics of particle
interactions and of the frictional behavior of the contacts. Internal geometric
constraints are imposed by the particles' shapes, in particular, by the surface
curvatures of the particles at their points of contact. Moreover, the stiffness
relationship is incrementally non-linear, and even small assemblies require the
analysis of multiple stiffness branches, with each branch region being a
pointed convex cone in displacement-space. These aspects of the particle-level
stiffness relationship gives rise to three types of micro-scale failure:
neutral equilibrium, bifurcation and path instability, and instability of
equilibrium. These three pathologies are defined in the context of four types
of displacement constraints, which can be readily analyzed with certain
generalized inverses. That is, instability and non-uniqueness are investigated
in the presence of kinematic constraints. Bifurcation paths can be either
stable or unstable, as determined with the Hill-Bazant-Petryk criterion.
Examples of simple granular systems of three, sixteen, and sixty four disks are
analyzed. With each system, multiple contacts were assumed to be at the
friction limit. Even with these small systems, micro-scale failure is expressed
in many different forms, with some systems having hundreds of micro-scale
failure modes. The examples suggest that micro-scale failure is pervasive
within granular materials, with particle arrangements being in a nearly
continual state of instability