pith. sign in

arxiv: 2601.00313 · v2 · pith:JTQ5QXI6new · submitted 2026-01-01 · ❄️ cond-mat.soft

Wrinkles, rucks, and folds formed in a heavy sheet on a frictional surface

Keisuke Yoshida , Hirofumi Wada This is my paper

Pith reviewed 2026-05-16 18:18 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords elastic sheetswrinklesrucksfoldsfrictiongravitybucklingmorphology
0
0 comments X

The pith

Lifting the center of a heavy elastic sheet produces wrinkles then rucks and folds, with a universal wrinkling threshold set by elasticity and gravity alone when friction is absent.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the shapes that develop when the center of a gravity-loaded elastic sheet is slowly lifted from a rigid supporting plane. Without friction the response depends only on elasticity and gravity, producing a fixed number of wrinkles at an onset displacement that grows linearly with sheet thickness. Friction is incorporated through one extra nondimensional parameter that measures frictional force relative to the elastic-gravitational restoring forces. The same sequence of axisymmetric uplift, wrinkles, global rucks, and folds appears consistently in experiments, finite-element simulations, and Föppl-von Kármán theory. The result supplies a compact route to controlling sheet shapes by tuning friction and gravity.

Core claim

Lifting the center of a gravity-loaded elastic sheet on a rigid surface drives a reproducible sequence of morphologies: an initial axisymmetric uplift, a finite number of wrinkles, system-spanning rucks produced by global buckling, and folded states that form when rucks collapse upon unloading at larger lifts. In the frictionless limit elasticity and gravity alone fix the wrinkling threshold, holding the wrinkle number constant while the critical displacement scales linearly with thickness. Friction is captured by a single nondimensional parameter that compares frictional forces to the combined elastic-gravitational forces.

What carries the argument

The nondimensional frictional parameter that compares frictional forces to elastic-gravitational forces, which extends the frictionless universal wrinkling threshold to frictional contact.

Load-bearing premise

A single nondimensional parameter fully describes the frictional contact without requiring local friction details or loading-history dependence.

What would settle it

Measuring a non-linear dependence of onset displacement on thickness, or a thickness-dependent wrinkle number, in a frictionless lifting experiment with sheets of several thicknesses would disprove the claimed universal threshold.

Figures

Figures reproduced from arXiv: 2601.00313 by Hirofumi Wada, Keisuke Yoshida.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Halloween ghost formed by draping a soft, thin fabric over a convex object. (b) Schematic of the experimental [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Lifting force and lifted radius vs. indentation height. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dimensionless (a) lifted radius [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The profile of the azimuthally averaged stress [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 8
Figure 8. Figure 8: Qualitatively, this can be understood as follows: [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Displacement field in the contact region for (i) small indentation height ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The number of wrinkles and the critical displacement [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Azimuthally averaged in-plane stress profiles obtained [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The number and critical displacement of wrinkles [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Relationships among the number of wrinkles [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Evolution of the deformation from (i) the seven-fold symmetric wrinkle pattern to (ii) a distorted, less-ordered pattern, [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Critical displacement of global buckling. The results [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. (a) Phase diagram classifying the observed wrinkling [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. (a) Force–displacement curve for the one cyclic test. We changed the maximum indentation height, [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
read the original abstract

Soft elastic sheets resting on rigid surfaces develop wrinkles, rucks, and folds due to the combined influence of elasticity, gravity, and contact interactions. Despite their ubiquity, the principles governing their morphology and transitions remain unclear. We introduce a minimal experiment in which the center of a gravity-loaded sheet is gradually lifted from the supporting plane. This operation generates a clear sequence of shapes: an axisymmetric uplift, a finite number of wrinkles, system-spanning rucks produced by global buckling, and folded states that can arise from ruck collapse upon unloading at larger lifts. Combining experiments, finite-element simulations, and F\"oppl-von K\'arm\'an theory, we establish a unified physical picture of this morphology sequence. In the frictionless case, elasticity and gravity alone govern the response, leading to a universal wrinkling threshold: the wrinkle number is fixed and the onset displacement scales linearly with the sheet thickness. With interfacial friction, the wrinkled state is described by introducing an additional nondimensional parameter that compares frictional and elastic-gravitational forces. These results suggest a simple route to programmable sheet morphogenesis via friction and gravity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript examines wrinkles, rucks, and folds in a heavy elastic sheet on a frictional surface by lifting its center. Combining experiments, finite-element simulations, and Föppl-von Kármán theory, it establishes a morphology sequence and scaling laws. In the frictionless case, a universal wrinkling threshold is identified with fixed wrinkle number and onset displacement scaling linearly with thickness. Friction is modeled by an additional nondimensional parameter comparing frictional and elastic-gravitational forces.

Significance. This provides a unified picture for gravity- and friction-driven sheet instabilities, with the parameter-free universal threshold in the frictionless limit being a notable result from nondimensionalization of the governing equations. The multi-method approach strengthens the findings, and the frictional parameter offers a route to programmable morphogenesis. The work is significant for soft matter physics and has potential applications in material design.

minor comments (3)
  1. The abstract mentions 'system-spanning rucks produced by global buckling', but a brief definition of ruck versus wrinkle would aid readers unfamiliar with the terminology.
  2. The experimental setup details, such as sheet material properties and surface preparation, should be more explicitly stated to allow replication.
  3. The comparison between theory, simulation, and experiment for the frictional case would be clearer with overlaid data points and a legend specifying the value of the frictional parameter used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, including the recognition of the universal wrinkling threshold in the frictionless limit and the utility of the frictional parameter. We appreciate the recommendation for minor revision and will prepare a revised version incorporating any editorial suggestions.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation rests on nondimensionalization of the standard Föppl-von Kármán equations augmented by gravity, which produces the claimed universal wrinkling threshold (onset displacement scaling linearly with thickness and fixed wrinkle number) as a direct consequence of the single remaining length scale set by bending-gravity balance. This scaling is independent of any fitted parameters internal to the target morphology. The frictional extension is introduced via an explicit nondimensional ratio of frictional to elastic-gravitational forces defined from force balance. The overall picture is corroborated by separate experiments and finite-element simulations rather than by self-referential fitting or self-citation chains that would collapse the central result onto its own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The analysis rests on the standard assumptions of Föppl-von Kármán thin-plate theory plus a simple frictional force model; no new entities are postulated and the frictional parameter is derived from dimensional comparison rather than fitted ad hoc.

free parameters (1)
  • frictional nondimensional parameter
    Compares frictional force scale to elastic-gravitational force scale; introduced to collapse data across different friction levels.
axioms (2)
  • standard math Föppl-von Kármán plate equations govern the sheet deformation
    Invoked to obtain analytic thresholds for wrinkling onset and ruck formation.
  • domain assumption Friction can be represented by a single effective nondimensional group
    Used to unify wrinkled states across different surface conditions.

pith-pipeline@v0.9.0 · 5500 in / 1378 out tokens · 97927 ms · 2026-05-16T18:18:40.191073+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages

  1. [1]

    Cerda, L

    E. Cerda, L. Mahadevan, and J. M. Pasini, The elements of draping, Proc. Natl. Acad. Sci. USA101, 1806 (2004)

    work page 2004

  2. [2]

    Chen and Y.-Y

    J.-S. Chen and Y.-Y. Fang, Non-axisymmetric warping of a heavy circular plate on a flexible ring support, Int. J. Solids Struct.47, 2767 (2010)

    work page 2010

  3. [3]

    Boedec and J

    G. Boedec and J. Deschamps, Disk wrinkling under grav- ity, Int. J. Solids Struct.233, 111214 (2021)

    work page 2021

  4. [4]

    Vandeparre, M

    H. Vandeparre, M. Pi˜ neirua, F. Brau, B. Roman, J. Bico, C. Gay, W. Bao, C. N. Lau, P. M. Reis, and P. Damman, Wrinkling hierarchy in constrained thin sheets from sus- pended graphene to curtains, Phys. Rev. Lett.106, 224301 (2011)

    work page 2011

  5. [5]

    Taffetani, F

    M. Taffetani, F. Box, A. Neveu, and D. Vella, Limitations of curvature-induced rigidity: How a curved strip buckles under gravity, Europhys. Lett.127, 14001 (2019)

    work page 2019

  6. [6]

    J. D. Johnston, J. R. Blandino, and K. C. McEvoy, Analytical and experimental characterization of gravity- induced deformations in subscale gossamer structures, J. Spacecr. Rockets43, 762 (2006)

    work page 2006

  7. [7]

    D. A. Spencer, L. Johnson, and A. C. Long, Solar sailing technology challenges, Aerosp. Sci. Technol.93, 105276 (2019)

    work page 2019

  8. [8]

    Karner and A

    G. Karner and A. Watts, Gravity anomalies and flexure of the lithosphere at mountain ranges, J. Geophys. Res.: Solid Earth88, 10449 (1983)

    work page 1983

  9. [9]

    Mahadevan, R

    L. Mahadevan, R. Bendick, and H. Liang, Why subduc- tion zones are curved, Tectonics29(2010)

    work page 2010

  10. [10]

    Proti` ere, C

    S. Proti` ere, C. Josserand, J. M. Aristoff, H. A. Stone, and M. Abkarian, Sinking a granular raft, Phys. Rev. Lett.118, 108001 (2017)

    work page 2017

  11. [11]

    Cerda and L

    E. Cerda and L. Mahadevan, Geometry and physics of wrinkling, Phys.Rev. Lett.90, 074302 (2003)

    work page 2003

  12. [12]

    E. P. Chan, E. J. Smith, R. C. Hayward, and A. J. Crosby, Surface wrinkles for smart adhesion, Adv. Mater. 20, 711 (2008)

    work page 2008

  13. [13]

    J. Y. Chung, A. J. Nolte, and C. M. Stafford, Surface wrinkling: a versatile platform for measuring thin-film properties, Adv. Mater.23, 349 (2011)

    work page 2011

  14. [14]

    Yamaguchi, S

    T. Yamaguchi, S. Ohmata, and M. Doi, Regular to chaotic transition of stick–slip motion in sliding friction of an adhesive gel-sheet, J. Phys. Condens. Matter21, 205105 (2009)

    work page 2009

  15. [15]

    Aoyanagi, J

    Y. Aoyanagi, J. Hure, J. Bico, and B. Roman, Random blisters on stickers: metrology through defects, Soft Mat- ter6, 5720 (2010)

    work page 2010

  16. [16]

    Chawla and D

    A. Chawla and D. Kumar, Geometry-induced friction at a soft interface, Proc. Natl. Acad. Sci. USA121, e2320068121 (2024)

    work page 2024

  17. [17]

    R. H. Plaut, S. Suherman, D. A. Dillard, B. E. Williams, and L. T. Watson, Deflections and buckling of a bent elastica in contact with a flat surface, Int. J. Solids Struct. 36, 1209 (1999)

    work page 1999

  18. [18]

    Roman and A

    B. Roman and A. Pocheau, Postbuckling of bilaterally constrained rectangular thin plates, J. Mech. Phys. Solids 50, 2379 (2002)

    work page 2002

  19. [19]

    Stoop, F

    N. Stoop, F. K. Wittel, and H. J. Herrmann, Morpho- logical phases of crumpled wire, Phys. Rev. Lett.101, 094101 (2008)

    work page 2008

  20. [20]

    Liu and J.-S

    C.-W. Liu and J.-S. Chen, Effect of coulomb friction on the deformation of an elastica constrained in a straight channel with clearance, Eur. J. Mech. A-Solid.39, 50 (2013)

    work page 2013

  21. [21]

    Alben, Packing of elastic rings with friction, P

    S. Alben, Packing of elastic rings with friction, P. R. Soc. A478, 20210742 (2022)

    work page 2022

  22. [22]

    Deboeuf, S

    S. Deboeuf, S. Proti` ere, and E. Katzav, Yin-yang spiral- ing transition of a confined buckled elastic sheet, Phys. Rev. Res.6, 013100 (2024)

    work page 2024

  23. [23]

    Mahadevan and J

    L. Mahadevan and J. B. Keller, Coiling of flexible ropes, Proc. R. Soc. A: Math. Phys. Eng. Sci.452, 1679 (1996)

    work page 1996

  24. [24]

    Habibi, N

    M. Habibi, N. Ribe, and D. Bonn, Coiling of elastic ropes, Phys. Rev. Lett.99, 154302 (2007)

    work page 2007

  25. [25]

    Vella, A

    D. Vella, A. Boudaoud, and M. Adda-Bedia, Statics and inertial dynamics of a ruck in a rug, Phys. Rev. Lett. 103, 174301 (2009)

    work page 2009

  26. [26]

    J. M. Kolinski, P. Aussillous, and L. Mahadevan, Shape and motion of a ruck in a rug, Phys. Rev. Lett.103, 174302 (2009)

    work page 2009

  27. [27]

    T. G. Sano, T. Yamaguchi, and H. Wada, Slip morphol- ogy of elastic strips on frictional rigid substrates, Phys. Rev. Lett.118, 178001 (2017)

    work page 2017

  28. [28]

    Grandgeorge, T

    P. Grandgeorge, T. G. Sano, and P. M. Reis, An elastic rod in frictional contact with a rigid cylinder, J. Mech. Phys. Solids164, 104885 (2022)

    work page 2022

  29. [29]

    Tani and H

    M. Tani and H. Wada, How a soft rod wraps around a rotating cylinder, Phys. Rev. Lett.132, 058204 (2024)

    work page 2024

  30. [30]

    G. K. Curtis, I. M. Griffiths, and D. Vella, Bridging a gap: A heavy elastica between point supports, Int. J. Solids Struct.326, 113702 (2026)

    work page 2026

  31. [31]

    Audoly and Y

    B. Audoly and Y. Pomeau,Elasticity and geometry: from hair curls to the non-linear response of shells(Oxford university press, 2010)

    work page 2010

  32. [32]

    T. A. Witten, Stress focusing in elastic sheets, Rev. Mod. Phys.79, 643 (2007)

    work page 2007

  33. [33]

    J. Hure, B. Roman, and J. Bico, Wrapping an adhesive sphere with an elastic sheet, Phys. Rev. Lett.106, 174301 (2011)

    work page 2011

  34. [34]

    J. Hure, B. Roman, and J. Bico, Stamping and wrinkling of elastic plates, Phys. Rev. Lett.109, 054302 (2012)

    work page 2012

  35. [35]

    Suzanne, J

    T. Suzanne, J. Deschamps, M. Georgelin, and G. Boedec, Indentation of an elastic disk on a circular supporting ring, Phys. Rev. E106, 065002 (2022)

    work page 2022

  36. [36]

    Stein-Montalvo, A

    L. Stein-Montalvo, A. Guerra, K. Almeida, O. Kodio, and D. P. Holmes, Wrinkling and developable cones in centrally confined sheets, Phys. Rev. E108, 035002 18 (2023)

    work page 2023

  37. [37]

    Krieg, G

    M. Krieg, G. Fl¨ aschner, D. Alsteens, B. M. Gaub, W. H. Roos, G. J. Wuite, H. E. Gaub, C. Gerber, Y. F. Dufrˆ ene, and D. J. M¨ uller, Atomic force microscopy- based mechanobiology, Nat. Rev. Phys.1, 41 (2019)

    work page 2019

  38. [38]

    Akinwande, C

    D. Akinwande, C. J. Brennan, J. S. Bunch, P. Egberts, J. R. Felts, H. Gao, R. Huang, J.-S. Kim, T. Li, Y. Li, et al., A review on mechanics and mechanical properties of 2d materials—graphene and beyond, Extreme Mech. Lett.13, 42 (2017)

    work page 2017

  39. [39]

    Chopin, D

    J. Chopin, D. Vella, and A. Boudaoud, The liquid blister test, Proc. R. Soc. A: Math. Phys. Eng. Sci.464, 2887 (2008)

    work page 2008

  40. [40]

    D. P. Holmes and A. J. Crosby, Draping films: A wrinkle to fold transition, Phys. Rev. Lett.105, 038303 (2010)

    work page 2010

  41. [41]

    Vella and B

    D. Vella and B. Davidovitch, Regimes of wrinkling in an indented floating elastic sheet, Phys. Rev. E98, 013003 (2018)

    work page 2018

  42. [42]

    Dannenberg, Measurement of adhesion by a blister method, J

    H. Dannenberg, Measurement of adhesion by a blister method, J. Appl. Polym. Sci.5, 125 (1961)

    work page 1961

  43. [43]

    Z. Dai, Y. Hou, D. A. Sanchez, G. Wang, C. J. Bren- nan, Z. Zhang, L. Liu, and N. Lu, Interface-governed deformation of nanobubbles and nanotents formed by two-dimensional materials, Phys. Rev. Lett.121, 266101 (2018)

    work page 2018

  44. [44]

    Z. Dai, D. A. Sanchez, C. J. Brennan, and N. Lu, Radial buckle delamination around 2d material tents, J. Mech. Phys. Solids137, 103843 (2020)

    work page 2020

  45. [45]

    V. L. Popov,Contact mechanics and friction(Springer, 2010)

    work page 2010

  46. [46]

    K. L. Johnson,Contact mechanics(Cambridge university press, 1987)

    work page 1987

  47. [47]

    Timoshenko and S

    S. Timoshenko and S. Woinowsky-Krieger,Theory of Plates and Shells, Engineering mechanics series (McGraw-Hill, 1959)

    work page 1959

  48. [48]

    D. A. Dillard, B. Mukherjee, P. Karnal, R. C. Batra, and J. Frechette, A review of winkler’s foundation and its pro- found influence on adhesion and soft matter applications, Soft matter14, 3669 (2018)

    work page 2018

  49. [49]

    F. Box, D. Vella, R. W. Style, and J. A. Neufeld, Indenta- tion of a floating elastic sheet: geometry versus applied tension, Proc. R. Soc. A: Math. Phys. Eng. Sci.473, 20170335 (2017)

    work page 2017

  50. [50]

    Wang, A critical review of the heavy elastica, Int

    C. Wang, A critical review of the heavy elastica, Int. J. Mech. Sci.28, 549 (1986)

    work page 1986

  51. [51]

    Vella and B

    D. Vella and B. Davidovitch, Indentation metrology of clamped, ultra-thin elastic sheets, Soft Matter13, 2264 (2017)

    work page 2017

  52. [52]

    R. C. Benson, Plate tenting with a one-sided constraint, J. Appl. Mech.58, 484 (1991)

    work page 1991

  53. [53]

    E. H. Mansfield,The bending and stretching of plates, 2nd ed. (Cambridge University Press, 1989)

    work page 1989

  54. [54]

    Dai and N

    Z. Dai and N. Lu, Poking and bulging of suspended thin sheets: Slippage, instabilities, and metrology, J. Mech. Phys. Solids149, 104320 (2021)

    work page 2021

  55. [55]

    Syst` emes, Abaqus version 6.6 documentation (2006)

    D. Syst` emes, Abaqus version 6.6 documentation (2006)

    work page 2006

  56. [56]

    Baumberger and C

    T. Baumberger and C. Caroli, Solid friction from stick– slip down to pinning and aging, Adv. Phys.55, 279 (2006)

    work page 2006

  57. [57]

    Timoshenko and J

    S. Timoshenko and J. Goodier,Theory of Elasticity, En- gineering mechanics series (McGraw-Hill, 1969)

    work page 1969

  58. [58]

    Davidovitch, R

    B. Davidovitch, R. D. Schroll, D. Vella, M. Adda-Bedia, and E. A. Cerda, Prototypical model for tensional wrin- kling in thin sheets, Proc. Natl. Acad. Sci. USA108, 18227 (2011)

    work page 2011

  59. [59]

    Vella, Buffering by buckling as a route for elastic de- formation, Nat

    D. Vella, Buffering by buckling as a route for elastic de- formation, Nat. Rev. Phys.1, 425 (2019)

    work page 2019

  60. [60]

    Domokos, W

    G. Domokos, W. Fraser, and I. Szeber´ enyi, Symmetry- breaking bifurcations of the uplifted elastic strip, Physica D185, 67 (2003)

    work page 2003

  61. [61]

    D´ emery, B

    V. D´ emery, B. Davidovitch, and C. D. Santangelo, Me- chanics of large folds in thin interfacial films, Phys. Rev. E90, 042401 (2014)

    work page 2014

  62. [62]

    D. L. Turcotte and G. Schubert,Geodynamics(Cam- bridge university press, 2002)

    work page 2002

  63. [63]

    Mahadevan and S

    L. Mahadevan and S. Rica, Self-organized origami, Sci- ence307, 1740 (2005)

    work page 2005

  64. [64]

    Felton, M

    S. Felton, M. Tolley, E. Demaine, D. Rus, and R. Wood, A method for building self-folding machines, Science345, 644 (2014)

    work page 2014

  65. [65]

    substantial slip

    T.-H. Kim, D.-Y. Lee, and J.-H. Han, Construction of metre-scale foldable space shelter based on gravity-driven self-assembling origami, Sci. Rep.15, 19615 (2025). Supplemental Material: Wrinkles, rucks and folds formed in a heavy sheet on a frictional surface Keisuke Yoshida 1, 2∗ and Hirofumi Wada 1 1Department of Physical Sciences, Ritsumeikan Universi...

    work page 2025

  66. [66]

    Vella and B

    D. Vella and B. Davidovitch, Indentation metrology of clamped, ultra-thin elastic sheets, Soft Matter 13, 2264 (2017)

    work page 2017

  67. [67]

    E. H. Mansfield, The bending and stretching of plates , 2nd ed. (Cambridge University Press, 1989)

    work page 1989

  68. [68]

    Chopin, D

    J. Chopin, D. Vella, and A. Boudaoud, The liquid blister test, Proc. R. Soc. A: Math. Phys. Eng. Sci. 464, 2887 (2008)

    work page 2008

  69. [69]

    Timoshenko and J

    S. Timoshenko and J. Goodier, Theory of Elasticity , Engineering mechanics series (McGraw-Hill, 1969). 7 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.1 1 2 1 1 1 (a) (b) FIG. S3. Relationship between T ≡ σrr(R) and the static friction coefficient µ obtained from FES. The elastic sheet parameter...

    work page 1969