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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
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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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Fig. 1: Average component of grain displacement parallel to the shear band plane (normalised by the maximum displacement at the considered strain interval) for systems A (a), B (b) and C (c) at strain intervals from the start to the end of each test: first interval (black), when minimum cut first predicts shear band location (blue), just prior to peak stress (green), and final when shear band is fully developed (red). Particle positions are measured normal to the shear band plane, then normalised to lie in the range [0,1], to permit comparison between systems.

Fig. 2: Particle rotation field and the minimum cut from the start (column (a)) to the end (column (h)) of loading history, for systems A (row A), B (row B) and C (row C). Each pair of images in columns (a)-(h) shows the sample at the beginning of the strain interval, with grain colours representing information across the strain interval. Increasing values of rotations (left image of each cell) are coloured from cool dark blue (lowest) to hot red (highest). In the minimum cuts (right image of each cell), green particles identify source/sink nodes; grains are coloured by their location on the source (red) or sink (blue) side of the minimum cut. Strain intervals: for A [0.0, 0.001]% (a), [0.4, 0.401]% (b), [0.8, 0.801]% (c), [3.2, 3.201]% (d), [5.2, 5.201]% (e), [7.6, 7.601]% (f), [9.6, 9.601]% (g), [12.0, 12.01]% (h), peak stress is 4.4% strain; for B [0, 0.176]% (a), [0.176, 0.914]% (b), [0.914, 1.82]% (c), [2.75, 3.65]% (d), [8.2, 9.11]% (e), [9.11, 10.02]% (f), [10.02, 10.94]%, (g), [10.94, 11.85]% (h), peak stress is 3.65% strain; for C [0, 1]% (a), [1, 2]% (b), [2, 3.5]% (c), [3.5, 5]% (d), [5, 7]% (e), [7, 9]% (f), [9, 12]% (g), [12, 17.5]% (h), peak stress is 3.5%-7% strain.

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Shear bands as bottlenecks in force transmission — Supplementary (a)

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Fig. 3: Two systems undergoing localised (a,b) versus diffuse (c,d) failure. Residual displacement (a) (i.e, the difference between the actual displacement and that dictated by the uniform strain) in the persistent shear banding regime, and the corresponding minimum cuts accumulated over many stages of this failure regime (b). Minimum cuts accumulated over many stages of the final cycle of the pure cyclic shear test: source and sink nodes are assigned to the left and right boundaries (c) versus top and bottom boundaries (d).

[2] Tordesillas A., Lin Q., Zhang J., Behringer R. P. and Shi J., Journal of the Mechanics and Physics of Solids, 59 (2011) 265. [3] Ren J., Dijksman J. A. and Behringer R. P., Physical Review Letters, 110 (2013) 018302. [4] Walker D. M., Tordesillas A., Ren J., Dijksman J. A. and Behringer R. P., Europhysics Letters, 107 (2014) 18005.

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