Abstract
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.
Article PDF
Similar content being viewed by others
References
Goldhirsch I.: Stress, stress asymmetry and couple stress: from discrete particles to continuous fields. Granul. Mat. 12(3), 239–252 (2010)
Frenkel D., Smit B.: Understanding Molecular Simulation, 1st edn. Academic Press, San Diego (1996)
Cundall P.A., Strack O.D.L.: A discrete numerical model for granular assemblies. Geotechnique 29(4765), 47–65 (1979)
Luding S.: Cohesive frictional powders: contact models for tension. Granul. Mat. 10(4), 235–246 (2008)
Monaghan J.J.: Smoothed particle hydrodynamics. Rep. Prog. Phys. 68(8), 1703–1759 (2005)
Luding S., Alonso-Marroquín F.: The critical-state yield stress (termination locus) of adhesive powders from a single numerical experiment. Granul. Mat. 13(2), 109–119 (2011)
Irving J.H., Kirkwood J.G.: The statistical mechanical theory of transport processes. J. Chem. Phys. 18, 817–829 (1950)
Todd B.D., Evans D.J., Daivis P.J.: Pressure tensor for inhomogeneous fluids. Phys. Rev. E 52(2), 1627–1638 (1995)
Babic M.: Average balance equations for granular materials. Int. J. Eng. Sci. 35(5), 523–548 (1997)
Morland L.W.: Flow of viscous fluid through a porous deformable matrix. Surv. Geophys. 13, 209–268 (1992)
Silbert L.E., Ertas D., Grest G.S., Halsey D., Levine T.C., Plimpton S.J.: Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E. 64, 051302 (2001)
Weinhart, T., Thornton, A.R., Luding, S., Bokhove, O.: Closure relations for shallow granular flows from particle simulations. Granul. Mat. (2011, submitted)
Acknowledgments
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.
Open Access
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Weinhart, T., Thornton, A.R., Luding, S. et al. From discrete particles to continuum fields near a boundary. Granular Matter 14, 289–294 (2012). https://doi.org/10.1007/s10035-012-0317-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10035-012-0317-4