Stress response function of a two-dimensional ordered packing of frictional beads

Abstract

We study the stress profile of an ordered two-dimensional packingof beads in response to the application of a vertical overloadlocalized at its top surface. Disorder is introduced through theCoulombic friction between the grains which gives someindeterminacy and allows the choice of one constrained randomnumber per grain in the calculation of the contact forces. Theso-called “multi-agent” technique we use lets us deal withsystems as large as 1000 × 1000 grains. We show that theaverage response profile has a double-peaked structure. At largedepth z, the position of these peaks grows with cz, whiletheir widths scales like √Dz. c and D are analogousto “propagation” and “diffusion” coefficients. Their valuesdepend on that of the friction coefficient μ. At small μ,we get c0 − c ∝ μ and D ∝ μβ, with β ∼ 2.5, which means that the peaks get closer and wider as thedisorder gets larger. This behavior is qualitatively what waspredicted in a model where a stochastic relation between thestress components is assumed.