Stress propagation through frictionless granular material

We examine the network of forces to be expected in a static assembly of hard, frictionless spherical beads of random sizes, such as a colloidal glass. Such an assembly is minimally connected: the ratio of constraint equations to contact forces approaches unity for a large assembly. However, the bead positions in a finite subregion of the assembly are underdetermined. Thus to maintain equilibrium, half of the exterior contact forces are determined by the other half. We argue that the transmission of force may be regarded as unidirectional, in contrast to the transmission of force in an elastic material. Specializing to sequentially deposited beads, we show that forces on a given buried bead can be uniquely specified in terms of forces involving more recently added beads. We derive equations for the transmission of stress averaged over scales much larger than a single bead. This derivation requires the ansatz that statistical fluctuations of the forces are independent of fluctuations of the contact geometry. Under this ansatz, the $d(d+1\mathrm{})/2\mathrm{}$-component stress field can be expressed in terms of a d-component vector field. The procedure may be generalized to nonsequential packings. In two dimensions, the stress propagates according to a wave equation, as postulated in recent work elsewhere. We demonstrate similar wavelike propagation in higher dimensions, assuming that the packing geometry has uniaxial symmetry. In macroscopic granular materials we argue that our approach may be useful even though grains have friction and are not packed sequentially.