- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Assur graphs, marginally jammed packings, and reconfigurable metamaterials

Jose Ortiz-Tavarez, Ethan Stanifer, and Xiaoming Mao

  • Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA

PDF

Phys. Rev. Research 5, L042001 – Published 2 October, 2023

DOI: https://doi.org/10.1103/PhysRevResearch.5.L042001

Abstract

Isostatic frames are mechanical networks that are simultaneously rigid and stress-free. Isostaticity itself is a powerful concept in understanding phase transitions in soft matter and designing mechanical metamaterials. We show that the contact network of marginally jammed packings approaches not only isostaticity but minimal isostaticity in the thermodynamic limit. We define in turn this minimal isostaticity and describe how global nonlocal mechanical responses are hallmarks of minimally isostatic graphs (MIGs) known previously as Assur graphs. By using the related concept of Assur decomposition, which we generalize for periodic boundary conditions, we not only assess our claim about jammed packings but we also offer a new design principle for mechanical metamaterials in which motion and stress can propagate in reconfigurable pathways, while rigidity of the entire structure is maintained. We also briefly note an apparent relationship between fully repulsive interactions and the emergence of MIGs at the unjamming point.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (48)

  1. S. Zhang, L. Zhang, M. Bouzid, D. Z. Rocklin, E. Del Gado, and X. Mao, Correlated Rigidity Percolation and Colloidal Gels, Phys. Rev. Lett. 123, 058001 (2019).
  2. J. Colombo and E. Del Gado, Stress localization, stiffening, and yielding in a model colloidal gel, J. Rheol. 58, 1089 (2014).
  3. C. S. O'Hern, L. E. Silbert, A. J. Liu, and S. R. Nagel, Jamming at zero temperature and zero applied stress: The epitome of disorder, Phys. Rev. E 68, 011306 (2003).
  4. D. Bi, J. Lopez, J. M. Schwarz, and M. L. Manning, A density-independent rigidity transition in biological tissues, Nat. Phys. 11, 1074 (2015).
  5. T. C. Lubensky, C. L. Kane, X. Mao, A. Souslov, and K. Sun, Phonons and elasticity in critically coordinated lattices, Rep. Prog. Phys. 78, 073901 (2015).
  6. X. Mao and T. C. Lubensky, Maxwell lattices and topological mechanics, Annu. Rev. Condens. Matter Phys. 9, 413 (2018).
  7. G. Laman, On graphs and rigidity of plane skeletal structures, J. Eng. Math. 4, 331 (1970).
  8. L. Asimow and B. Roth, The rigidity of graphs, Trans. Am. Math. Soc. 245, 279 (1978).
  9. S. Alexander, Amorphous solids: their structure, lattice dynamics and elasticity, Phys. Rep. 296, 65 (1998).
  10. M. van Hecke, Jamming of soft particles: geometry, mechanics, scaling and isostaticity, J. Phys.: Condens. Matter 22, 033101 (2010).
  11. A. J. Liu and S. R. Nagel, The jamming transition and the marginally jammed solid, Annu. Rev. Condens. Matter Phys. 1, 347 (2010).
  12. C. P. Goodrich, A. J. Liu, and J. P. Sethna, Scaling ansatz for the jamming transition, Proc. Natl. Acad. Sci. USA 113, 9745 (2016).
  13. D. J. Jacobs and M. F. Thorpe, Generic Rigidity Percolation: The Pebble Game, Phys. Rev. Lett. 75, 4051 (1995).
  14. A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, A linear programming algorithm to test for jamming in hard-sphere packings, J. Comput. Phys. 197, 139 (2004).
  15. M. Mailman, C. F. Schreck, C. S. O'Hern, and B. Chakraborty, Jamming in Systems Composed of Frictionless Ellipse-Shaped Particles, Phys. Rev. Lett. 102, 255501 (2009).
  16. C. P. Broedersz, X. Mao, T. C. Lubensky, and F. C. MacKintosh, Criticality and isostaticity in fibre networks, Nat. Phys. 7, 983 (2011).
  17. W. G. Ellenbroek and X. Mao, Rigidity percolation on the square lattice, Europhys. Lett. 96, 54002 (2011).
  18. D. Bi, J. Zhang, B. Chakraborty, and R. P. Behringer, Jamming by shear, Nature (London) 480, 355 (2011).
  19. L. Yan and M. Wyart, Evolution of Covalent Networks under Cooling: Contrasting the Rigidity Window and Jamming Scenarios, Phys. Rev. Lett. 113, 215504 (2014).
  20. W. Ellenbroek, V. Hagh, A. Kumar, M. Thorpe, and M. Hecke, Rigidity loss in disordered systems: Three scenarios, Phys. Rev. Lett. 114, 135501 (2015).
  21. S. Henkes, D. A. Quint, Y. Fily, and J. M. Schwarz, Rigid Cluster Decomposition Reveals Criticality in Frictional Jamming, Phys. Rev. Lett. 116, 028301 (2016).
  22. R. P. Behringer and B. Chakraborty, The physics of jamming for granular materials: a review, Rep. Prog. Phys. 82, 012601 (2019).
  23. E. Hahn and O. Shai, The unique engineering properties of assur groups/graphs, assur kinematic chains, baranov trusses and parallel robots, in International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (ASME Press, New York, NY, 2016), Vol. 50169, p. V05BT07A074.
  24. O. Shai, A. Sljoka, and W. Whiteley, Directed graphs, decompositions, and spatial linkages, Discr. Appl. Math. 161, 3028 (2013).
  25. J. C. Maxwell, On the calculation of the equilibrium and stiffness of frames, London, Edinburgh, Dublin Philos. Mag. J. Sci. 27, 294 (1864).
  26. R. Connelly and W. Whiteley, Second-order rigidity and prestress stability for tensegrity frameworks, SIAM J. Discrete Math. 9, 453 (1996).
  27. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.5.L042001 for proofs of the original theorems, further explanations of the algorithm used, and other details.
  28. L. V. Assur and I. I. Artobolevskij, Issledovanie Ploskih Steržnevyh Mehanizmov s Nizšimi Parami s Točki Zreniâ ih Struktury i Klassifikacii (Izdatel'stvo Akademii Nauk SSSR, Moscow, 1952).
  29. A. Sljoka, O. Shai, and W. Whiteley, Checking mobility and decomposition of linkages via pebble game algorithm, in International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 6: 35th Mechanisms and Robotics Conference, Parts A and B of International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (ASME Press, New York, NY, 2011), pp. 493–502.
  30. R. Penne, Isostatic bar and joint frameworks in the plane with irreducible pure conditions, Disc. Appl. Math. 55, 37 (1994).
  31. S. Pellegrino and C. R. Calladine, Matrix analysis of statically and kinematically indeterminate frameworks, Int. J. Solids Struct. 22, 409 (1986).
  32. A. Lee and I. Streinu, Pebble game algorithms and sparse graphs, Disc. Math. 308, 1425 (2008).
  33. E. Ross, Geometric and combinatorial rigidity of periodic frameworks as graphs on the torus, Ph.D. thesis, Graduate Program in the Department of Mathematics and Statistics, York University Toronto, Ontario (2011).
  34. E. Bitzek, P. Koskinen, F. Gähler, M. Moseler, and P. Gumbsch, Structural Relaxation Made Simple, Phys. Rev. Lett. 97, 170201 (2006).
  35. B. Roth and W. Whiteley, Tensegrity frameworks, Trans. Am. Math. Soc. 265, 419 (1981).
  36. B. Servatius, O. Shai, and W. Whiteley, Geometric properties of assur graphs, Eur. J. Combinat. 31, 1105 (2010).
  37. J. Paulose, B. G.-G. Chen, and V. Vitelli, Topological modes bound to dislocations in mechanical metamaterials, Nat. Phys. 11, 153 (2015).
  38. D. Rocklin, S. Zhou, K. Sun, and X. Mao, Transformable topological mechanical metamaterials, Nat. Commun. 8, 14201 (2017).
  39. D. Rocklin, V. Vitelli, and X. Mao, Folding mechanisms at finite temperature, arXiv:1802.02704.
  40. K. Sun and X. Mao, Continuum Theory for Topological Edge Soft Modes, Phys. Rev. Lett. 124, 207601 (2020).
  41. H. Xiu, H. Liu, A. Poli, G. Wan, K. Sun, E. M. Arruda, X. Mao, and Z. Chen, Topological transformability and reprogrammability of multistable mechanical metamaterials, Proc. Natl. Acad. Sci. USA 119, e2211725119 (2022).
  42. S. Zhang, E. Stanifer, V. V. Vasisht, L. Zhang, E. Del Gado, and X. Mao, Prestressed elasticity of amorphous solids, Phys. Rev. Res. 4, 043181 (2022).
  43. Topological Synthesis of All 2D Mechanisms Through Assur Graphs, Volume 2: 34th Annual Mechanisms and Robotics Conference, Parts A and B of International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 08 (ASME Press, New York, NY, 2010).
  44. C. Toninelli, G. Biroli, and D. S. Fisher, Jamming Percolation and Glass Transitions in Lattice Models, Phys. Rev. Lett. 96, 035702 (2006).
  45. J. M. Schwarz, A. J. Liu, and L. Q. Chayes, The onset of jamming as the sudden emergence of an infinite k-core cluster, Europhys. Lett. 73, 560 (2006).
  46. L. Zhang, D. Z. Rocklin, B. G.-G. Chen, and X. Mao, Rigidity percolation by next-nearest-neighbor bonds on generic and regular isostatic lattices, Phys. Rev. E 91, 032124 (2015).
  47. J. H. Lopez, L. Cao, and J. M. Schwarz, Jamming graphs: A local approach to global mechanical rigidity, Phys. Rev. E 88, 062130 (2013).
  48. V. F. Hagh, E. I. Corwin, K. Stephenson, and M. Thorpe, Jamming in perspective, arXiv:1803.03869.

Outline

Information

Phys. Rev. Research 5, L042001– Published 2 October, 2023
  • Received 22 December 2022
  • Accepted 2 August 2023
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation