A closure relation for the stress in static granular media under gravity

Unlike fluids, no established principles govern the stress in static granular media under gravity, though they are encountered widely in nature and industry. Experiments have show that the stress depends strongly on how the grain assembly is created and the nature of confining boundaries. Non-trivial stress variation even in simple geometries have posed long-standing challenges to continuum modelling. We have studied the problem computationally by creating gravity-deposited grain packings and studying force transmission in the grain contact network. We determine the paths of load transmission averaged over an ensemble of realizations, or 'force lines', and show that they encode preparation history. We hypothesize that the weight of each particle is transmitted by biased random walks to the particles below, and validate the hypothesis by showing excellent agreement between the ensemble-averaged random walks and force lines. This identification leads to a closure relation for the static stress, that yields a parabolic partial differential equation for the vertical normal stress. The predictions of the model are shown to compare very well with the stress obtained from particle dynamics simulations.