Response of a hexagonal granular packing under a localized external force: exact results

Abstract

Abstract. -We study the response of a two-dimensional hexagonal packing of rigid, frictionless spherical grains due to a vertically downward point force on a single grain at the top layer. We use a statistical approach, where each configuration of the contact forces is equally likely. We show that this problem is equivalent to a correlated q-model. We find that the response displays two peaks which lie precisely on the downward lattice directions emanating from the point of application of the force. With increasing depth, the magnitude of the peaks decreases, and a central peak develops. On the bottom of the pile, only the middle peak persists. The response of different system sizes exhibits self-similarity.Force transmissions in (static) granular packings have attracted a lot of attention in recent years [1][2][3][4][5][6][7][8][9][10][11][12][13]. Granular packings are assemblies of macroscopic particles that interact only via mechanical repulsion effected through physical contacts. Experimental and numerical studies of these systems have identified two main characteristics. First, large fluctuations are found to occur in the magnitudes of inter-grain forces, implying that the probability distribution of the force magnitudes is rather broad [4]. Secondly, the average propagation of forcesstudied via the response to a single external force -is strongly dependent on the underlying contact geometry [3,[11][12][13].The available theoretical models capture either one or the other of these two aspects. The scalar q-model [5] reproduces reasonably well the observed force distribution, but yields diffusive propagation of forces, in conflict with experiments [11,12]. Continuum elastic and elasto-plastic theories [6] predict responses in qualitative agreement with experiments [7-10], but they provide a description only at the average macroscopic level. More ad-hoc "stressonly" models [2] include structural randomness, but its consequences on the distribution of forces are unclear. In other words, an approach that produces both realistic fluctuations and propagation of forces in granular materials from the same set of fundamental principles is still called for.A simple conjecture, which could provide such a fundamental principle for all problems of granular statics, has been put forward by Edwards years ago [14,15]. The idea is to consider all "jammed" configurations equally probable. A priori, there is no justification for such an c EDP Sciences

Sensitivity of the stress response function to packing preparation

“…From all these data, as in 2D, we conclude the existence of a macroscopic anisotropy induced by the pile preparation. Similar conclusion could be drawn for 3D response functions experiments [28,29] or simulations and calculations performed on highly symmetrical crystalline packing [13,30,31,32,33].…”

Response of a hexagonal granular packing under a localized external force

“…Following a similar path, we studied the response of a two-dimensional hexagonal packing of rigid, frictionless spherical grains (see Fig. 1) to a pointlike external force F , and found that it is concentrated mainly on the two lattice directions emanating from the point of application of the force (Ostojic & Panja 2005), in agreement with experiments (Geng 2001). Our definition of the response of the packing is…”

“…Nevertheless, one has to be careful: notice that the G i,j 's are differences of the physical contact forces and thus they are allowed to become negative in magnitude. In < F 0 appear elsewhere [18,19]) This arrangement reduces the uniform measure over E to the uniform measure on S ′ , which is a (N + 1)p-dimensional polyhedron.…”

Elasticity from the Force Network Ensemble in Granular Media

“…In Refs. [26,27], we examined this packing under zero vertical and large horizontal pressure. While this situation allows an instructive comparison between the force ensemble and the q-model [28], physically it corresponds to a singular limit, where there is no relevant scale for the applied overload [30].…”