Socolar, JES; Schaeffer, DG; Claudin, P, *Directed force chain networks and stress response in static granular materials*,
European Physical Journal E, vol. 7 no. 4
(2002),
pp. 353-370 [0201242] .
**Abstract:**

*A theory of stress fields in two-dimensional granular materials based on directed force chain networks is presented. A general Boltzmann equation for the densities of force chains in different directions is proposed and a complete solution is obtained for a special case in which chains lie along a discrete set of directions. The analysis and results demonstrate the necessity of including nonlinear terms in the Boltzmann equation. A line of nontrivial fixed-point solutions is shown to govern the properties of large systems. In the vicinity of a generic fixed point, the response to a localized load shows a crossover from a single, centered peak at intermediate depths to two propagating peaks at large depths that broaden diffusively.*