From discrete particles to continuum fields near a boundary
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An expression for the stress tensor near an external boundary of a discrete mechanical system is derived explicitly in terms of the constituents’ degrees of freedom and interaction forces. Starting point is the exact and general coarse graining formulation presented by Goldhirsch (Granul Mat 12(3):239–252, 2010), which is consistent with the continuum equations everywhere but does not account for boundaries. Our extension accounts for the boundary interaction forces in a self-consistent way and thus allows the construction of continuous stress fields that obey the macroscopic conservation laws even within one coarse-graining width of the boundary. The resolution and shape of the coarse-graining function used in the formulation can be chosen freely, such that both microscopic and macroscopic effects can be studied. The method does not require temporal averaging and thus can be used to investigate time-dependent flows as well as static or steady situations. Finally, the fore-mentioned continuous field can be used to define ‘fuzzy’ (very rough) boundaries. Discrete particle simulations are presented in which the novel boundary treatment is exemplified, including chute flow over a base with roughness greater than one particle diameter.
KeywordsCoarse graining Averaging Boundary treatment DPM (DEM) Discrete mechanical systems Homogenisation Stress Continuum mechanics Granular systems
The authors would like to thank the Institute for Mechanics, Process, and Control, Twente (IMPACT) and the NWOSTW VICI grant 10828 for financial support, and Remco Hartkamp and Dinant Krijgsman for fruitful discussions. The method presented will benefit our research on “Polydispersed Granular Flows through Inclined Channels—Influence of Particle Characteristics, Channel Rotation and Geometry” funded by STW.
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
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