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3 of the Let- ter with the inclusion of additional results for other strain intervals in the loading history. 3. Minimum cuts in the presence versus absence of shear bands. – Given the ease of visualising pat- terns in 2D versus 3D, we demonstrate in Fig. 3 the un- ambiguous association between the shear band and the.
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epl draft

Shear bands as bottlenecks in force transmission — Supplementary Antoinette Tordesillas12 , Sebastian Pucilowski1 , Steven Tobin1 , ` 45 , Gioacchino Viggiani45 , Matthew R. Kuhn3 , Edward Ando 6 Andrew Druckrey and Khalid Alshibli6 1 2 3 4 5 6

Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia School of Earth Sciences, University of Melbourne, Parkville, Victoria 3010, Australia Department of Civil Engineering, School of Engineering, University of Portland, OR 97203 USA Universit´e Grenoble Alpes, 3SR, F-38000 Grenoble, France CNRS, 3SR, F-38000 Grenoble, France Department of Civil and Environmental Engineering, University of Tennessee, Knoxville, TN 37996 USA

PACS PACS PACS

81.05.Rm – Porous materials; granular materials 7.55.de – Optimization 05.10.-a – Computational methods in statistical physics and nonlinear dynamics

Abstract – Supplementary

1. Average grain displacement profile. – To assess the predictive power of the minimum cut, we probed the evolution of the shear band in the early stages of loading using the raw data on grain kinematics and knowledge of the location and plane of the persistent shear band. No evidence of the shear band manifests until around peak stress ratio: compare the linear average displacement profiles in the initial stages of loading prior to peak stress versus the step change in the profile for the last stage of each test, indicative of the shear band, in Fig. 1.

subject the sample to uniform deformation at all stages of the test [3, 4]: thus, shear bands do not form in this sample. In both systems, minimum cuts are computed for successive stages of loading, using the method demonstrated and discussed in the Letter. In the system that undergoes localised failure [1, 2], the accumulated minimum cuts (Fig. 3 (b)) localise along the shear band. In the system that undergoes diffuse failure (i.e., failure in the absence of shear bands) [3, 4], the accumulated minimum cuts do not localise and remain diffuse (Fig. 3 (c,d)), regardless of which walls are chosen to be the source and sink.

2. Particle rotations and minimum cuts. – Fig. 2 shows visualisations of the particle rotation field and the minimum cut from the start (column (a)) to the end (colAcknowledgements. – We thank Robert P. umn (h)) of loading history, for systems A (row A), B (row Behringer, Jie Zhang, Jie Ren and Joshua A. Dijksman B) and C (row C). This figure extends Fig. 3 of the Let- for sharing their data for this study. ter with the inclusion of additional results for other strain This work was supported by ARC DP120104759 intervals in the loading history. & US ARO W911NF-11-1-0175 for Antoinette Tordesillas, Sebastian Pucilowski & Steven Tobin, and 3. Minimum cuts in the presence versus absence US NSF CMMI-1362510 & ONR N00014-11-1-0691 for of shear bands. – Given the ease of visualising patAndrew Druckrey & Khalid Alshibli. terns in 2D versus 3D, we demonstrate in Fig. 3 the unambiguous association between the shear band and the localisation of minimum cuts for two systems in 2D that were recently reported in the published literature. The REFERENCES first system underwent localised failure in the presence of a single persistent shear band (Fig. 3 (a)) [1,2]. The second [1] Zhang J., Majmudar T. S., Tordesillas A. and Behringer R. P., Granular Matter, 12 (2010) 159. system is a pure cyclic shear test specifically designed to p-1

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