Stability, bifurcation, and softening in discrete systems: A conceptual approach for granular materials

Matrix stiffness expressions are derived for the particle movements in an assembly of rigid granules having compliant contacts. The derivations include stiffness terms that arise from the particle shapes at their contacts. These geometric stiffness terms may become significant during granular failure. The geometric stiffness must be added to the mechanical stiffnesses of the contacts to produce the complete stiffness. With frictional contacts, this stiffness expression is incrementally nonlinear, having multiple loading branches. To aid the study of material behavior, a modified stiffness is derived for isolated granular clusters that are considered detached from the rest of a granular body. Criteria are presented for bifurcation, instability, and softening of such isolated and discrete granular sub-regions. Examples show that instability and softening can result entirely from the geometric terms in the matrix stiffness.