Mean Stress Tensor of Discrete Particle Systems in Submerged Conditions

The mean stress tensor is essential to investigate the dynamics of granular material. In this paper, we use Hamilton's principle of least action to derive the averaged stress tensor of discrete granular assemblies subjected to hydraulic force fields, as well as rigorous conditions for a proper definition of the Representative Volume Element (RVE). The main goal behind our efforts is to upscale particle physics into a sound stress tensor for systems involving the complex interaction between grains and water. We identify the contributions from the unbalanced forces, hydraulic forces, gravity, external forces, and particle fluctuation to the mean stress tensor. In doing so, it is convenient to separate the influence of different force fields when the granular system is subjected to complex environments, e.g., subaqueous conditions. The obtained formula is then validated by triaxial test simulations of dry and saturated granular systems using the Discrete Element Method (DEM) and the Lattice-Boltzmann Method (LBM). The results show that the deduced formula can accurately calculate the stress tensor of discrete assemblies with various body-force fields. We used validated DEM-LBM simulations of submerged granular column collapses to explore the physics happening at the grain scale with this mathematical formalism and showcase its potential. We provide a new perspective based on the granular assembly scale to pursue the fluid-solid interaction. Due to the importance of stress analysis in the constitutive modelling of granular materials, this work could help to better obtain the stress-strain relationship of saturated or submerged granular systems.