Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Achieving adjustable elasticity with non-affine to affine transition

Abstract

For various engineering and industrial applications it is desirable to realize mechanical systems with broadly adjustable elasticity to respond flexibly to the external environment. Here we discover a topology-correlated transition between affine and non-affine regimes in elasticity in both two- and three-dimensional packing-derived networks. Based on this transition, we numerically design and experimentally realize multifunctional systems with adjustable elasticity. Within one system, we achieve solid-like affine response, liquid-like non-affine response and a continuous tunability in between. Moreover, the system also exhibits a broadly tunable Poisson’s ratio from positive to negative values, which is of practical interest for energy absorption and for fracture-resistant materials. Our study reveals a fundamental connection between elasticity and network topology, and demonstrates its practical potential for designing mechanical systems and metamaterials.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Non-affine to affine transition in packing-derived networks.
Fig. 2: Understanding the variation of moduli at the single-particle level.
Fig. 3: The numerical design of a network with both affine and non-affine tunability.
Fig. 4: Experimental realization of affine and non-affine tunability.
Fig. 5: Extending the 2D results into three dimensions.

Data availability

All the raw data of the figures presented in this manuscript can be obtained by accessing the Open Science Framework (https://doi.org/10.17605/OSF.IO/7EQ5Z) or directly visiting https://osf.io/7eq5z/ (ref. 47).

Code availability

All custom computer code and algorithms used to generate the results reported in the paper are available upon request.

References

  1. 1.

    Kadic, M., Bückmann, T., Schittny, R. & Wegener, M. Metamaterials beyond electromagnetism. Rep. Prog. Phys. 76, 126501 (2013).

    Article  Google Scholar 

  2. 2.

    Berger, J. B., Wadley, H. N. G. & McMeeking, R. M. Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness. Nature 543, 533–537 (2017).

    CAS  Article  Google Scholar 

  3. 3.

    Rocks, J. W. et al. Designing allostery-inspired response in mechanical networks. Proc. Natl Acad. Sci. USA 114, 2520–2525 (2017).

    CAS  Article  Google Scholar 

  4. 4.

    Reid, D. R. et al. Auxetic metamaterials from disordered networks. Proc. Natl Acad. Sci. USA 115, 1384–1390 (2018).

    Article  CAS  Google Scholar 

  5. 5.

    Nicolaou, Z. G. & Motter, A. E. Mechanical metamaterials with negative compressibility transitions. Nat. Mater. 11, 608–613 (2012).

    CAS  Article  Google Scholar 

  6. 6.

    Coulais, C., Sounas, D. & Alù, A. Static non-reciprocity in mechanical metamaterials. Nature 542, 461–464 (2017).

    CAS  Article  Google Scholar 

  7. 7.

    Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2014).

    CAS  Article  Google Scholar 

  8. 8.

    Zheng, X. et al. Ultralight, ultrastiff mechanical metamaterials. Science 344, 1373–1377 (2014).

    CAS  Article  Google Scholar 

  9. 9.

    Coulais, C., Sabbadini, A., Vink, F. & van Hecke, M. Multi-step self-guided pathways for shape-changing metamaterials. Nature 561, 512–515 (2018).

    CAS  Article  Google Scholar 

  10. 10.

    Goodrich, C. P., Liu, A. J. & Nagel, S. R. The principle of independent bond-level response: tuning by pruning to exploit disorder for global behavior. Phys. Rev. Lett. 114, 225501 (2015).

    Article  CAS  Google Scholar 

  11. 11.

    Florijn, B., Coulais, C. & van Hecke, M. Programmable mechanical metamaterials. Phys. Rev. Lett. 113, 175503 (2014).

    Article  CAS  Google Scholar 

  12. 12.

    Majmudar, T. S., Sperl, M., Luding, S. & Behringer, R. P. Jamming transition in granular systems. Phys. Rev. Lett. 98, 058001 (2007).

    CAS  Article  Google Scholar 

  13. 13.

    Dauchot, O., Marty, G. & Biroli, G. Dynamical heterogeneity close to the jamming transition in a sheared granular material. Phys. Rev. Lett. 95, 265701 (2005).

    CAS  Article  Google Scholar 

  14. 14.

    Keys, A. S., Abate, A. R., Glotzer, S. C., & Durian, D. J. Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material. Nat. Phys. 4, 260–264 (2007).

    Article  CAS  Google Scholar 

  15. 15.

    Alexander, S. Amorphous solids: their structure, lattice dynamics and elasticity. Phys. Rep. 296, 65–236 (1998).

    CAS  Article  Google Scholar 

  16. 16.

    Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C. & Levine, D. Geometry of frictionless and frictional sphere packings. Phys. Rev. E 65, 031304 (2002).

    Article  CAS  Google Scholar 

  17. 17.

    Oswald, L., Grosser, S., Smith, D. M. & Kas, J. A. Jamming transitions in cancer. J. Phys. D 50, 483001 (2017).

    Article  CAS  Google Scholar 

  18. 18.

    Mongera, A. et al. A fluid-to-solid jamming transition underlies vertebrate body axis elongation. Nature 561, 401–405 (2018).

    CAS  Article  Google Scholar 

  19. 19.

    Liu, A. J. & Nagel, S. R. Jamming is not just cool any more. Nature 396, 21–22 (1998).

    CAS  Article  Google Scholar 

  20. 20.

    Olsson, P. & Teitel, S. Critical scaling of shear viscosity at the jamming transition. Phys. Rev. Lett. 99, 178001 (2007).

    Article  CAS  Google Scholar 

  21. 21.

    Head, D. A. Critical scaling and aging in cooling systems near the jamming transition. Phys. Rev. Lett. 102, 138001 (2009).

    Article  CAS  Google Scholar 

  22. 22.

    Somfai, E., Roux, J. N., Snoeijer, J. H., Van Hecke, M. & Van Saarloos, W. Elastic wave propagation in confined granular systems. Phys. Rev. E 72, 021301 (2005).

    Article  CAS  Google Scholar 

  23. 23.

    Makse, H. A., Gland, N., Johnson, D. L. & Schwartz, L. M. Why effective medium theory fails in granular materials. Phys. Rev. Lett. 83, 5070 (1999).

    CAS  Article  Google Scholar 

  24. 24.

    Makse, H. A., Johnson, D. L. & Schwartz, L. M. Packing of compressible granular materials. Phys. Rev. Lett. 84, 4160 (2000).

    CAS  Article  Google Scholar 

  25. 25.

    Zhang, Z. X. et al. Thermal vestige of the zero-temperature jamming transition. Nature 459, 230–233 (2009).

    CAS  Article  Google Scholar 

  26. 26.

    Weitz, D. Packing in the spheres. Science 303, 968–969 (2004).

    CAS  Article  Google Scholar 

  27. 27.

    Tkachenko, A. V. & Witten, T. A. Stress propagation through frictionless granular material. Phys. Rev. E 60, 687 (1999).

    CAS  Article  Google Scholar 

  28. 28.

    Ellenbroek, W. G., van Hecke, M. & van Saarloos, W. Jammed frictionless disks: connecting local and global response. Phys. Rev. E 80, 061307 (2009).

    Article  CAS  Google Scholar 

  29. 29.

    Wyart, M., Nagel, S. R. & Witten, T. A. Geometric origin of excess low-frequency vibrational modes in weakly connected amorphous solids. Europhys. Lett. 72, 486 (2005).

    CAS  Article  Google Scholar 

  30. 30.

    Wyart, M., Silbert, L. E., Nagel, S. R. & Witten, T. A. Effects of compression on the vibrational modes of marginally jammed solids. Phys. Rev. E 72, 051306 (2005).

    Article  CAS  Google Scholar 

  31. 31.

    Ellenbroek, W. G., Somfai, E., van Hecke, M. & Van Saarloos, W. Critical scaling in linear response of frictionless granular packings near jamming. Phys. Rev. Lett. 97, 258001 (2006).

    Article  CAS  Google Scholar 

  32. 32.

    Donev, A., Torquato, S. & Stillinger, F. H. Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings. Phys. Rev. E 71, 011105 (2005).

    Article  CAS  Google Scholar 

  33. 33.

    Silbert, L. E., Liu, A. J. & Nagel, S. R. Vibrations and diverging length scales near the unjamming transition. Phys. Rev. Lett. 95, 098301 (2005).

    Article  CAS  Google Scholar 

  34. 34.

    Zhao, C., Tian, K. & Xu, N. New jamming scenario: from marginal jamming to deep jamming. Phys. Rev. Lett. 106, 125503 (2011).

    Article  CAS  Google Scholar 

  35. 35.

    Mao, X., Xu, N. & Lubensky, T. C. Soft modes and elasticity of nearly isostatic lattices: randomness and dissipation. Phys. Rev. Lett. 104, 085504 (2010).

    Article  CAS  Google Scholar 

  36. 36.

    Xu, N., Vitelli, V., Liu, A. J. & Nagel, S. R. Anharmonic and quasi-localized vibrations in jammed solids—modes for mechanical failure. Europhys. Lett. 90, 56001 (2010).

    Article  CAS  Google Scholar 

  37. 37.

    Liu, A. J. & Nagel, S. R. The jamming transition and the marginally jammed solid. Annu. Rev. Condens. Matter Phys. 1, 347–369 (2010).

    Article  Google Scholar 

  38. 38.

    van Hecke, M. Jamming of soft particles: geometry, mechanics, scaling and isostaticity. J. Phys. Condens. Matter 22, 033101 (2010).

    Article  CAS  Google Scholar 

  39. 39.

    Calladine, C. R. Buckminster Fuller tensegrity structures and Clerk Maxwell rules for the construction of stiff frames. Int. J. Solids Struct. 14, 161–172 (1978).

    Article  Google Scholar 

  40. 40.

    Wyart, M., Liang, H., Kabla, A. & Mahadevan, L. Elasticity of floppy and stiff random networks. Phys. Rev. Lett. 101, 215501 (2008).

    CAS  Article  Google Scholar 

  41. 41.

    Lubensky, T. C., Kane, C. L., Mao, X., Souslov, A. & Sun, K. Phonons and elasticity in critically coordinated lattices. Rep. Prog. Phys. 78, 073901 (2015).

    CAS  Article  Google Scholar 

  42. 42.

    Trudeau, R. J. Introduction to Graph Theory 64–116 (Dover, 1993).

  43. 43.

    Nakamura, N. Geometry, Topology and Physics 67–91 (Institute of Physics Publishing, 2008).

  44. 44.

    Giménez, O. & Noy, M. Asymptotic enumeration and limit laws of planar graphs. J. Am. Math. Soc. 22, 309–329 (2009).

    Article  Google Scholar 

  45. 45.

    Ellenbroek, W. G., Zeravcic, Z., Van Saarloos, W. & Van Hecke, M. Non-affine response: jammed packings vs spring networks. Europhys. Lett. 87, 34004 (2009).

    Article  CAS  Google Scholar 

  46. 46.

    Li, S. et al. Liquid-induced topological transformations of cellular microstructures. Nature 592, 386–391 (2021).

    CAS  Article  Google Scholar 

  47. 47.

    Shen, X. et al. Achieving Adjustable Elasticity with Non-affine to Affine Transition https://doi.org/10.17605/OSF.IO/7EQ5Z (OSF, 2021).

Download references

Acknowledgements

The experiments were performed at The Chinese University of Hong Kong, and we acknowledge the computational support from the Beijing Computational Science Research Center. L.X. acknowledges financial support from NSFC-12074325, Guangdong Basic and Applied Basic Research Fund 2019A1515011171, GRF-14306518, CRF-C6016-20G, CRF-C1018-17G, CUHK United College Lee Hysan Foundation Research Grant and Endowment Fund Research Grant, CUHK direct grant 4053354. X.X. acknowledges financial support from NSFC 11974038 and U1930402. X.S. acknowledges financial support from Guangdong Basic and Applied Basic Research Foundation 2019A1515110211, and project funding by China Postdoctoral Science Foundation 2020M672824.

Author information

Affiliations

Authors

Contributions

X.S. and C.F. contributed equally to this research. L.X. conceived the research. X.S., C.F., J.H.Y.L., X.X. and L.X. designed the research. X.S. performed most of the theoretical and numerical analysis. C.F. performed most of the experiments. Z.J., S.T., H.S., H.T. and N.X. helped in the experiments or the simulations. X.S., C.F. and L.X. prepared the manuscript. L.X. and X.X. supervised the research.

Corresponding authors

Correspondence to Jack Hau Yung Lo or Xinliang Xu or Lei Xu.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review informationNature Materials thanks Larry Howell, Yang Jiao, Zachary Nicolaou and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–11 and discussion.

Supplementary Video 1

Producing negative Poisson’s ratio in our system.

Supplementary Video 2

Adding intersecting bonds does not change the Poisson’s ratio.

Supplementary Video 3

Illustration of bond removal and addition operations in our 2D spring network.

Supplementary Video 4

The comparison of internal strain fields between two 3D networks at z = 7.696 and z = 9.312.

Supplementary Video 5

The comparison of internal strain fields between two 3D networks at z = 9.312 and z = 10.432.

Supplementary Video 6

Illustration of 3D-printed detachable bonds.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shen, X., Fang, C., Jin, Z. et al. Achieving adjustable elasticity with non-affine to affine transition. Nat. Mater. (2021). https://doi.org/10.1038/s41563-021-01046-8

Download citation

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing