Stochastic Approach to the Solution of Boussinesq-Like Problems in Discrete Media

A vertical surface load acting on a half-space made of discrete and elastic particles is 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 from those predicted by the unique 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 the principal directions of the stress established by the continuum-based approach, the probability distribution functions of the extensive normal and shear stresses of particles are anticipated to be exponential and Laplace distributions, respectively. The extensive stress is the product of the volumetric average of the stress field within a region by the volume of that region. The parameters locating and scaling these probability distribution functions (PDFs) are such that the expected values of the extensive stresses match the solution to the corresponding boundary value problem: zero extensive shear stress and extensive normal stresses equal to the principal ones. The continuum-based approach is still needed to know the expected values, but this research article presents a powerful method for quantifying their expected variability. The theory 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 on small length scales (with particular applications for laboratory testing).