# Boussinesq-like problems in discrete media

## TL;DR

This paper investigates the relationship between discrete particle models and continuum solutions for Boussinesq-like problems, using statistical mechanics and numerical simulations to connect force chain distributions with classical stress fields.

## Contribution

It introduces a statistical mechanics framework to relate discrete force chain distributions to continuum stress solutions in Boussinesq-like problems, validated by extensive discrete element method simulations.

## Key findings

- Force chain stress distributions are exponential for normal components.
- Shear component distributions follow Laplace distributions.
- Parameters of distributions can be derived from continuum solutions.

## Abstract

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.