Stress Response Function to a two-dimensional ordered packing of frictional beads
Résumé
We study the stress profile of an ordered two-dimensional packing of beads in response to the application of a vertical overload localized at its top surface. Disorder is introduced through the Coulombic friction between the grains which gives some indeterminacy and allows the choice of one constrained random number per grain in the calculation of the contact forces. The so-called "multi-agent" technique we use lets us deal with systems as large as $1000 \times 1000$ grains. We show that the average response profile has a double-peaked structure. At large depth $z$, the position of these peaks grows with $cz$, while their widths scales like $\sqrt{Dz}$. $c$ and $D$ are analogous to "propagation" and "diffusion" coefficients. Their values depend on that of the friction coefficient $\mu$. At small $\mu$, we get $c_0 - c \propto \mu$ and $D \propto \mu^\beta$, with $\beta \sim 2.5$, which means that the peaks get closer and wider as the disorder gets larger. This behavior is qualitatively what was predicted in a model where a stochastic relation between the stress components is assumed.