Boussinesq-like problems in discrete media

Vertical loads acting on the surface of a half-space made of discrete and elastic particles are supported by a network of force chains that changes with the specific realization of the packing. These force chains can be transformed into equivalent stress fields, but the obtained values are usually different to those expected from the solution of the corresponding boundary value problem. In this research the relationship between discrete and continuum approaches to Boussinesq-like problems is explored in the light of classical statistical mechanics. In principal directions, the anticipated statistical distributions of the extensive stress (\textit{i.e.} the product of the stress by the volume) are exponential distributions for normal components and Laplace distributions for shear components. The parameters scaling these distributions can be obtained from the solutions provided by continuum approaches in most of the cases. This has been validated through massive numerical simulation with the discrete element method. These results could be of interest in highly fragmented, faulted or heterogeneous media or for small length scales.