  We investigate both numerically and analytically the effect of strong
disorder on the large scale properties of the hyperbolic equations for stresses
proposed in \protect\cite{bcc,wcc}. The physical mechanism that we model is the
local splitting of the force chains (the characteristics of the hyperbolic
equation) by packing defects. In analogy with the theory of light diffusion in
a turbid medium, we propose a Boltzmann-like equation to describe these
processes. We show that, for isotropic packings, the resulting large scale
effective equations for the stresses have exactly the same structure as those
of an elastic body, despite the fact that no displacement field needs to be
introduced at all. Correspondingly, the response function evolves from a two
peak structure at short scales to a broad hump at large scales. We find,
however, that the Poisson ratio is anomalously large and incompatible with
classical elasticity theory that requires the reference state to be
thermodynamically stable.
