In recent years the physics of granular materials has received growing interest [1]. One of the many interesting features of granulates is the stress distribution in static or quasi-static arrays. In contrast to a liquid, the pressure in a silo, filled with e.g. grains, is not increasing linearly with depth, but saturates at a certain value [2]. This is due to internal friction and due to arching, so that the walls of the silo carry a part of the materials' weight. In sandpiles no walls are present so that the situation may be different, i.e. the total weight of the pile has to be carried by the bottom. However, the distribution of forces under and also inside the pile is not yet completely understood. Experiments on rather large piles show that the normal force has a relative minimum under the top of the pile, the so-called dip [3, 4]. On a much smaller scale, the stress chains are observed, i.e. stresses are mainly transported along selected paths and the probability distribution of stress spans orders of magnitude [5, 6, 7].
One simple model pile is an array of rigid spheres, arranged on a diamond lattice, i.e. with four nearest neighbors each [8, 9]. The force under such a pile is constant in contrast to the experimental observations, and also periodic vacancies in such a configuration do not lead to a dip in the pressure at the bottom [10]. The variation of the size of some of the particles or an attractive force between the particles may lead to a non-constant force under the pile [11]. Continuum approaches [12, 13, 14, 15, 16] may lead to a dip in the vertical stress if the correct assumptions for the constitutive equations are chosen. Edwards [12] introduced the notion that a pressure minimum can result from compressive stresses aligning in fixed directions. Wittmer et al. [15, 16] embellished this idea recently with concrete calculations in agreement with the experimental data [4]. A lattice model based on a random opening of contacts [17] also shows the dip in average over many realizations.
In this study we focus on 2D-situations, with particles on an almost regular lattice, which we analyse using MD-simulations. The aim is to find the dip under conditions as simple as possible and to understand the stress networks and arches. We describe the simulation method used in Sec. ii and discuss the physics of particle contacts in Sec. iii. The results are presented in Sec. iv and are discussed in Sec. v.