A fundamental property of any material is its response to a
localized stress applied at a boundary. For granular materials
consisting of hard, cohesionless particles, not even the general
form of the stress response is known. Directed force chain networks
(DFCNs) provide a theoretical framework for addressing this issue,
and analysis of simplified DFCN models reveal both rich mathematical
structure and surprising properties. We review some basic elements
of DFCN models and present a class of homogeneous solutions for
cases in which force chains are restricted to lie on a discrete set
of directions.