Stresses in isostatic granular systems and emergence of force chains

TL;DR

This paper derives explicit hyperbolic stress equations for 2D isostatic granular systems, explaining the emergence of force chains and proposing a framework to extend insights to more realistic granular packings.

Contribution

It provides the first explicit hyperbolic stress equations in 2D isostatic systems and links these to the formation of force chains from first principles.

Findings

Stress equations are hyperbolic in 2D isostatic systems.

Force chains arise naturally as solutions to these equations.

Predicted force chain trajectories depend on local geometry.

Abstract

Progress is reported on several questions that bedevil understanding of granular systems: (i) are the stress equations elliptic, parabolic or hyperbolic? (ii) how can the often-observed force chains be predicted from a first-principles continuous theory? (iii) How to relate insight from isostatic systems to general packings? Explicit equations are derived for the stress components in two dimensions including the dependence on the local structure. The equations are shown to be hyperbolic and their general solutions, as well as the Green function, are found. It is shown that the solutions give rise to force chains and the explicit dependence of the force chains trajectories and magnitudes on the local geometry is predicted. Direct experimental tests of the predictions are proposed. Finally, a framework is proposed to relate the analysis to non-isostatic and more realistic granular assemblies.