Stress propagation through frictionless granular material

We examine the network of forces to be expected in a static assembly of har d, 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 m aterial. Specializing to sequentially deposited beads, we show that forces on a given buried bead can be uniquely specified in terms of forces involvi ng more recently added beads. We derive equations for the transmission of s tress averaged over scales much larger than a single bead. This derivation requires the ansatz that statistical fluctuations of the forces are indepen dent of fluctuations of the contact geometry. Under this ansatz, the d(d 1)/2-component stress field can be expressed in terms of a d-component vect or held. The procedure may be generalized to nonsequential packings. In two dimensions, the stress propagates according to a wave equation, as postula ted in recent work elsewhere. We demonstrate similar wavelike propagation i n higher dimensions, assuming that the packing geometry has uniaxial symmet ry. In macroscopic granular materials we argue that our approach may be use ful even though grains have friction and are not packed sequentially. [S106 3-651X(99)02007-3].