Towards an optimization theory for deforming dense granular materials: Minimum cost maximum flow solutions

Qun Lin; Antoinette Tordesillas

doi:10.3934/jimo.2014.10.337

Article Contents

2014, Volume 10, Issue 1: 337-362. Doi: 10.3934/jimo.2014.10.337

This issue Previous Article Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation Next Article

Towards an optimization theory for deforming dense granular materials: Minimum cost maximum flow solutions

Qun Lin^1, and
Antoinette Tordesillas^2,

1.
Department of Mathematics and Statistics, Curtin University, GPO Box U1987 Perth, Western Australia 6845
2.
Department of Mathematics and Statistics, University of Melbourne, Melbourne, Australia 3010

Received: February 2013

Revised: July 2013

Published: January 2014

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Abstract

We use concepts and techniques of network optimization theory to gain a better understanding of force transmission in dense granular materials. Specifically, we represent a deforming granular material over the different stages of a quasi-static biaxial compression test as a series of representative flow networks, and analyze force transmission through these networks. The forces in such a material are transmitted through the contacts between the constituent grains. As the sample deforms during the various stages of the biaxial test, these grains rearrange: while many contacts are preserved in this rearrangement process, some new contacts form and some old contacts break. We consider the maximum flow problem and the minimum cost maximum flow (MCMF) problem for the flow networks constructed from this evolving network of grain contacts. We identify the flow network bottleneck and establish the sufficient and necessary conditions for a minimum cut of the maximum flow problem to be unique. We also develop an algorithm to determine the MCMF pathway, i.e. a set of edges that always transmit non-zero flow in every solution of the MCMF problem. The bottlenecks of the flow networks develop in the locality of the persistent shear band, an intensively-studied phenomenon that has long been regarded as the signature failure microstructure for dense granular materials. The cooperative evolution of the most important structural building blocks for force transmission, i.e. the force chains and 3-cycles, is examined with respect to the MCMF pathways. We find that the majority of the particles in the major load-bearing columnar force chains and 3-cycles consistently participate in the MCMF pathways.

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Mathematics Subject Classification: Primary: 90B10, 90C35, 90C27; Secondary: 76T25.

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