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

## TL;DR

This paper develops a comprehensive stiffness model for granular materials, incorporating geometric effects and analyzing conditions for bifurcation, instability, and softening in discrete granular systems.

## Contribution

It introduces a detailed stiffness formulation including geometric effects and criteria for instability in granular assemblies, advancing understanding of granular failure modes.

## Key findings

- Geometric stiffness can cause instability and softening.
- Modified stiffness models for isolated granular clusters.
- Instability can arise solely from geometric effects.

## Abstract

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.

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Source: https://tomesphere.com/paper/1901.07340