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

– 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, let us deal with systems as large as 1000×1000 grains. We show that the average response profile has a double peaked structure. At large depth z, the position of these peaks grows like with cz, while their widths scales like √ Dz. c and D are analogous to ‘propagation’ and ‘diffusion’ coefficients. Their values depend on that of the friction coefficient μ. At small μ, we get c0 − c ∝ μ and D ∝ μ , with β ∼ 2.5, which means that the peaks get closer and wider as the disorder gets larger. This behavior is qualitatively that was predicted in a model where a stochastic relation between the stress components is assumed. The statics of granular materials is subject to an active research, see e.g. [1]. One of the main issue concerns the link between the distribution of stresses in a granular medium and its ‘past history’ which has induced, at the microscopic level, a certain texture to the packing. As a matter of fact, the mechanical properties of an assembly of grains depends on the way it has been prepared. A now famous example is that of the sandpile: when built from the source point of a hopper, the pressure profile at the bottom of a pile shows an unambiguous minimum right below its apex [2]. By contrast, this profile is almost flat with a slight hump when the pile has been prepared by successive horizontal layers [3]. The difference between the two sandpiles could be seen at the level of the grains whose contacts and forces are oriented in relation to the external solicitation they were applied: during the construction, grains tend to gain strong contacts – i.e. carrying a large force – in the direction of compression [4]. This strong texture can be visualized in photoelasticity experiments which show clear ‘force chains’ structures [5]. The pressure minimum at the bottom of a sandpile can then be interpreted as a screening effect of the weight of the grains by these chains.